diff --git a/pytorch_lightning/trainer/connectors/accelerator_connector.py b/pytorch_lightning/trainer/connectors/accelerator_connector.py
index 8271cf9bdc742..a80a470cb6f94 100644
--- a/pytorch_lightning/trainer/connectors/accelerator_connector.py
+++ b/pytorch_lightning/trainer/connectors/accelerator_connector.py
@@ -277,7 +277,8 @@ def _set_devices_if_none(self) -> None:
             self.devices = self.num_processes
 
     def _handle_accelerator_and_strategy(self) -> None:
-        if self.distributed_backend is not None and self.distributed_backend in list(DistributedType):
+        deprecated_types = [t for t in DistributedType if t not in (DistributedType.TPU_SPAWN, DistributedType.DDP_CPU)]
+        if self.distributed_backend is not None and self.distributed_backend in deprecated_types:
             rank_zero_deprecation(
                 f"Passing `Trainer(accelerator={self.distributed_backend!r})` has been deprecated"
                 f" in v1.5 and will be removed in v1.7. Use `Trainer(strategy={self.distributed_backend!r})` instead."
diff --git a/tests/accelerators/test_accelerator_connector.py b/tests/accelerators/test_accelerator_connector.py
index 7ad93b167794d..3d56a3a44ee9f 100644
--- a/tests/accelerators/test_accelerator_connector.py
+++ b/tests/accelerators/test_accelerator_connector.py
@@ -68,7 +68,8 @@ def test_accelerator_choice_ddp_cpu(tmpdir, num_processes: int, num_nodes: int):
 @mock.patch("torch.cuda.device_count", return_value=2)
 @mock.patch("torch.cuda.is_available", return_value=True)
 def test_accelerator_choice_ddp(cuda_available_mock, device_count_mock):
-    trainer = Trainer(fast_dev_run=True, accelerator="ddp", gpus=1)
+    with pytest.deprecated_call(match=r"accelerator='ddp'\)` has been deprecated"):
+        trainer = Trainer(fast_dev_run=True, accelerator="ddp", gpus=1)
     assert isinstance(trainer.accelerator, GPUAccelerator)
     assert isinstance(trainer.training_type_plugin, DDPPlugin)
     assert isinstance(trainer.training_type_plugin.cluster_environment, LightningEnvironment)
@@ -78,7 +79,8 @@ def test_accelerator_choice_ddp(cuda_available_mock, device_count_mock):
 @mock.patch("torch.cuda.device_count", return_value=2)
 @mock.patch("torch.cuda.is_available", return_value=True)
 def test_accelerator_choice_ddp_spawn(cuda_available_mock, device_count_mock):
-    trainer = Trainer(fast_dev_run=True, accelerator="ddp_spawn", gpus=1)
+    with pytest.deprecated_call(match=r"accelerator='ddp_spawn'\)` has been deprecated"):
+        trainer = Trainer(fast_dev_run=True, accelerator="ddp_spawn", gpus=1)
     assert isinstance(trainer.accelerator, GPUAccelerator)
     assert isinstance(trainer.training_type_plugin, DDPSpawnPlugin)
     assert isinstance(trainer.training_type_plugin.cluster_environment, LightningEnvironment)
@@ -105,7 +107,7 @@ def on_fit_start(self, trainer, pl_module):
             assert isinstance(trainer.training_type_plugin, DDPPlugin)
             assert isinstance(trainer.training_type_plugin.cluster_environment, SLURMEnvironment)
             assert trainer.training_type_plugin.cluster_environment.local_rank() == 1
-            assert trainer.training_type_plugin.task_idx == 1
+            assert trainer.training_type_plugin.local_rank == 1
             raise SystemExit()
 
     model = BoringModel()
@@ -137,7 +139,7 @@ def on_fit_start(self, trainer, pl_module):
             assert isinstance(trainer.training_type_plugin, DDP2Plugin)
             assert isinstance(trainer.training_type_plugin.cluster_environment, SLURMEnvironment)
             assert trainer.training_type_plugin.cluster_environment.local_rank() == 1
-            assert trainer.training_type_plugin.task_idx == 1
+            assert trainer.training_type_plugin.local_rank == 1
             raise SystemExit()
 
     model = BoringModel()
@@ -168,7 +170,7 @@ def on_fit_start(self, trainer, pl_module):
             assert isinstance(trainer.training_type_plugin, DDPPlugin)
             assert isinstance(trainer.training_type_plugin.cluster_environment, TorchElasticEnvironment)
             assert trainer.training_type_plugin.cluster_environment.local_rank() == 1
-            assert trainer.training_type_plugin.task_idx == 1
+            assert trainer.training_type_plugin.local_rank == 1
             raise SystemExit()
 
     model = BoringModel()
@@ -199,7 +201,7 @@ def on_fit_start(self, trainer, pl_module):
             assert isinstance(trainer.training_type_plugin, DDP2Plugin)
             assert isinstance(trainer.training_type_plugin.cluster_environment, TorchElasticEnvironment)
             assert trainer.training_type_plugin.cluster_environment.local_rank() == 1
-            assert trainer.training_type_plugin.task_idx == 1
+            assert trainer.training_type_plugin.local_rank == 1
             raise SystemExit()
 
     model = BoringModel()
@@ -221,7 +223,7 @@ def on_fit_start(self, trainer, pl_module):
             assert isinstance(trainer.training_type_plugin, DDPPlugin)
             assert isinstance(trainer.training_type_plugin.cluster_environment, TorchElasticEnvironment)
             assert trainer.training_type_plugin.cluster_environment.local_rank() == 1
-            assert trainer.training_type_plugin.task_idx == 1
+            assert trainer.training_type_plugin.local_rank == 1
             raise SystemExit()
 
     model = BoringModel()
@@ -252,7 +254,7 @@ def on_fit_start(self, trainer, pl_module):
             assert isinstance(trainer.training_type_plugin, DDPPlugin)
             assert isinstance(trainer.training_type_plugin.cluster_environment, KubeflowEnvironment)
             assert trainer.training_type_plugin.cluster_environment.local_rank() == 0
-            assert trainer.training_type_plugin.task_idx == 0
+            assert trainer.training_type_plugin.local_rank == 0
             raise SystemExit()
 
     model = BoringModel()
@@ -281,7 +283,7 @@ def on_fit_start(self, trainer, pl_module):
             assert isinstance(trainer.training_type_plugin, DDPPlugin)
             assert isinstance(trainer.training_type_plugin.cluster_environment, KubeflowEnvironment)
             assert trainer.training_type_plugin.cluster_environment.local_rank() == 0
-            assert trainer.training_type_plugin.task_idx == 0
+            assert trainer.training_type_plugin.local_rank == 0
             raise SystemExit()
 
     model = BoringModel()
@@ -311,7 +313,7 @@ def on_fit_start(self, trainer, pl_module):
             assert isinstance(trainer.accelerator, CPUAccelerator)
             assert isinstance(trainer.training_type_plugin, DDPPlugin)
             assert isinstance(trainer.training_type_plugin.cluster_environment, SLURMEnvironment)
-            assert trainer.training_type_plugin.task_idx == 0
+            assert trainer.training_type_plugin.local_rank == 0
             raise SystemExit()
 
     model = BoringModel()
@@ -435,7 +437,7 @@ class CB(Callback):
         def on_fit_start(self, trainer, pl_module):
             assert isinstance(trainer.accelerator, CPUAccelerator)
             assert isinstance(trainer.training_type_plugin, DDPPlugin)
-            assert trainer.training_type_plugin.task_idx == 0
+            assert trainer.training_type_plugin.local_rank == 0
             raise SystemExit()
 
     model = BoringModel()
@@ -463,10 +465,16 @@ def test_ipython_compatible_backend(*_):
 @pytest.mark.parametrize(["accelerator", "plugin"], [("ddp_spawn", "ddp_sharded"), (None, "ddp_sharded")])
 def test_plugin_accelerator_choice(accelerator: Optional[str], plugin: str):
     """Ensure that when a plugin and accelerator is passed in, that the plugin takes precedent."""
-    trainer = Trainer(accelerator=accelerator, plugins=plugin, num_processes=2)
+    if accelerator is None:
+        with pytest.deprecated_call(match="Passing .* `strategy` to the `plugins`"):
+            trainer = Trainer(accelerator=accelerator, plugins=plugin, num_processes=2)
+    else:
+        with pytest.deprecated_call(match=r"accelerator=.*\)` has been deprecated"):
+            trainer = Trainer(accelerator=accelerator, plugins=plugin, num_processes=2)
     assert isinstance(trainer.accelerator.training_type_plugin, DDPShardedPlugin)
 
-    trainer = Trainer(plugins=plugin, num_processes=2)
+    with pytest.deprecated_call(match="Passing .* `strategy` to the `plugins`"):
+        trainer = Trainer(plugins=plugin, num_processes=2)
     assert isinstance(trainer.accelerator.training_type_plugin, DDPShardedPlugin)
 
 
@@ -486,7 +494,8 @@ def test_plugin_accelerator_choice(accelerator: Optional[str], plugin: str):
 def test_accelerator_choice_multi_node_gpu(
     mock_is_available, mock_device_count, tmpdir, accelerator: str, plugin: ParallelPlugin, gpus: int
 ):
-    trainer = Trainer(accelerator=accelerator, default_root_dir=tmpdir, num_nodes=2, gpus=gpus)
+    with pytest.deprecated_call(match=r"accelerator=.*\)` has been deprecated"):
+        trainer = Trainer(accelerator=accelerator, default_root_dir=tmpdir, num_nodes=2, gpus=gpus)
     assert isinstance(trainer.training_type_plugin, plugin)
 
 
@@ -630,13 +639,16 @@ def test_unsupported_distrib_types_on_cpu(training_type):
 
 
 def test_accelerator_ddp_for_cpu(tmpdir):
-    trainer = Trainer(accelerator="ddp", num_processes=2)
+    with pytest.deprecated_call(match=r"accelerator='ddp'\)` has been deprecated"):
+        trainer = Trainer(accelerator="ddp", num_processes=2)
     assert isinstance(trainer.accelerator, CPUAccelerator)
     assert isinstance(trainer.training_type_plugin, DDPPlugin)
 
 
 def test_exception_when_strategy_used_with_accelerator():
-    with pytest.raises(MisconfigurationException, match="but have also passed"):
+    with pytest.raises(MisconfigurationException, match="but have also passed"), pytest.deprecated_call(
+        match=r"accelerator='ddp'\)` has been deprecated"
+    ):
         Trainer(accelerator="ddp", strategy="ddp_spawn")
 
 
@@ -762,7 +774,7 @@ def on_fit_start(self, trainer, pl_module):
             assert isinstance(trainer.training_type_plugin, DDPPlugin)
             assert isinstance(trainer.training_type_plugin.cluster_environment, SLURMEnvironment)
             assert trainer.training_type_plugin.cluster_environment.local_rank() == 1
-            assert trainer.training_type_plugin.task_idx == 1
+            assert trainer.training_type_plugin.local_rank == 1
             raise SystemExit()
 
     model = BoringModel()
@@ -794,7 +806,7 @@ def on_fit_start(self, trainer, pl_module):
             assert isinstance(trainer.training_type_plugin, DDP2Plugin)
             assert isinstance(trainer.training_type_plugin.cluster_environment, SLURMEnvironment)
             assert trainer.training_type_plugin.cluster_environment.local_rank() == 1
-            assert trainer.training_type_plugin.task_idx == 1
+            assert trainer.training_type_plugin.local_rank == 1
             raise SystemExit()
 
     model = BoringModel()
@@ -825,7 +837,7 @@ def on_fit_start(self, trainer, pl_module):
             assert isinstance(trainer.training_type_plugin, DDPPlugin)
             assert isinstance(trainer.training_type_plugin.cluster_environment, TorchElasticEnvironment)
             assert trainer.training_type_plugin.cluster_environment.local_rank() == 1
-            assert trainer.training_type_plugin.task_idx == 1
+            assert trainer.training_type_plugin.local_rank == 1
             raise SystemExit()
 
     model = BoringModel()
@@ -856,7 +868,7 @@ def on_fit_start(self, trainer, pl_module):
             assert isinstance(trainer.training_type_plugin, DDP2Plugin)
             assert isinstance(trainer.training_type_plugin.cluster_environment, TorchElasticEnvironment)
             assert trainer.training_type_plugin.cluster_environment.local_rank() == 1
-            assert trainer.training_type_plugin.task_idx == 1
+            assert trainer.training_type_plugin.local_rank == 1
             raise SystemExit()
 
     model = BoringModel()
@@ -878,7 +890,7 @@ def on_fit_start(self, trainer, pl_module):
             assert isinstance(trainer.training_type_plugin, DDPPlugin)
             assert isinstance(trainer.training_type_plugin.cluster_environment, TorchElasticEnvironment)
             assert trainer.training_type_plugin.cluster_environment.local_rank() == 1
-            assert trainer.training_type_plugin.task_idx == 1
+            assert trainer.training_type_plugin.local_rank == 1
             raise SystemExit()
 
     model = BoringModel()
@@ -909,7 +921,7 @@ def on_fit_start(self, trainer, pl_module):
             assert isinstance(trainer.training_type_plugin, DDPPlugin)
             assert isinstance(trainer.training_type_plugin.cluster_environment, KubeflowEnvironment)
             assert trainer.training_type_plugin.cluster_environment.local_rank() == 0
-            assert trainer.training_type_plugin.task_idx == 0
+            assert trainer.training_type_plugin.local_rank == 0
             raise SystemExit()
 
     model = BoringModel()
@@ -938,7 +950,7 @@ def on_fit_start(self, trainer, pl_module):
             assert isinstance(trainer.training_type_plugin, DDPPlugin)
             assert isinstance(trainer.training_type_plugin.cluster_environment, KubeflowEnvironment)
             assert trainer.training_type_plugin.cluster_environment.local_rank() == 0
-            assert trainer.training_type_plugin.task_idx == 0
+            assert trainer.training_type_plugin.local_rank == 0
             raise SystemExit()
 
     model = BoringModel()
@@ -968,7 +980,7 @@ def on_fit_start(self, trainer, pl_module):
             assert isinstance(trainer.accelerator, CPUAccelerator)
             assert isinstance(trainer.training_type_plugin, DDPPlugin)
             assert isinstance(trainer.training_type_plugin.cluster_environment, SLURMEnvironment)
-            assert trainer.training_type_plugin.task_idx == 0
+            assert trainer.training_type_plugin.local_rank == 0
             raise SystemExit()
 
     model = BoringModel()