DeepSpeed fp16 mixed precision converts weights to fp16 too
### Description & Motivation

```
Trainer(precision='16-mixed', strategy='deepspeed', ...)
```
The configuration above is using fp16 mode (instead of automatic mix-precision mode) in DeepSpeed. This is very misleading since fp16 mode is numerically unstable in many cases. 

Maybe lighting need a new precision option for DeepSpeed, or output a warning to inform users that they are in fact using fp16 mode.

### Pitch

_No response_

### Alternatives

_No response_

### Additional context

_No response_

cc @awaelchli @borda
diff --git a/src/lightning/fabric/strategies/deepspeed.py b/src/lightning/fabric/strategies/deepspeed.py
index d9e0b226f..0cb39db08 100644
--- a/src/lightning/fabric/strategies/deepspeed.py
+++ b/src/lightning/fabric/strategies/deepspeed.py
@@ -51,7 +51,7 @@ class DeepSpeedStrategy(DDPStrategy, _Sharded):
         accelerator: Optional[Accelerator] = None,
         zero_optimization: bool = True,
         stage: int = 2,
-        remote_device: str = "cpu",
+        remote_device: Optional[str] = None,
         offload_optimizer: bool = False,
         offload_parameters: bool = False,
         offload_params_device: str = "cpu",
@@ -112,7 +112,7 @@ class DeepSpeedStrategy(DDPStrategy, _Sharded):
                 1 is optimizer state partitioning, 2 is optimizer+gradient state partitioning,
                 3 is optimizer+gradient_parameter partitioning using the infinity engine.
 
-            remote_device: Device to instantiate the model on initially (``cpu`` or ``nvme``).
+            remote_device: Device to instantiate the model on initially (``cpu`` or ``nvme``). Defaults to GPU.
 
             offload_optimizer: Enable offloading optimizer memory and computation to CPU or NVMe
                 based on ``offload_optimizer_device``.
diff --git a/src/lightning/pytorch/strategies/deepspeed.py b/src/lightning/pytorch/strategies/deepspeed.py
index 3a7f1c6a3..cec7ec232 100644
--- a/src/lightning/pytorch/strategies/deepspeed.py
+++ b/src/lightning/pytorch/strategies/deepspeed.py
@@ -78,7 +78,7 @@ class DeepSpeedStrategy(DDPStrategy):
         accelerator: Optional["pl.accelerators.Accelerator"] = None,
         zero_optimization: bool = True,
         stage: int = 2,
-        remote_device: str = "cpu",
+        remote_device: Optional[str] = None,
         offload_optimizer: bool = False,
         offload_parameters: bool = False,
         offload_params_device: str = "cpu",
@@ -139,7 +139,7 @@ class DeepSpeedStrategy(DDPStrategy):
                 1 is optimizer state partitioning, 2 is optimizer+gradient state partitioning,
                 3 is optimizer+gradient_parameter partitioning using the infinity engine.
 
-            remote_device: Device to instantiate the model on initially (``cpu`` or ``nvme``).
+            remote_device: Device to instantiate the model on initially (``cpu`` or ``nvme``). Defaults to GPU.
 
             offload_optimizer: Enable offloading optimizer memory and computation to CPU or NVMe
                 based on ``offload_optimizer_device``.
['src/lightning/fabric/strategies/deepspeed.py', 'src/lightning/pytorch/strategies/deepspeed.py']