GH60942 Update docs and overload for Series.rename (#60975)

* Update docs and overload for Series.rename

* GH60942 Revert removal of overloads

* GH60942 Run pre-commit

* Update pandas/core/series.py

Co-authored-by: Matthew Roeschke <[email protected]>

---------

Co-authored-by: Matthew Roeschke <[email protected]>

Author: loicdiridollou

pandas/core/series.py

@@ -4651,7 +4651,7 @@ def rename(
         inplace: Literal[True],
         level: Level | None = ...,
         errors: IgnoreRaise = ...,
-    ) -> None: ...
+    ) -> Series | None: ...
 
     @overload
     def rename(
@@ -4665,18 +4665,6 @@ def rename(
         errors: IgnoreRaise = ...,
     ) -> Series: ...
 
-    @overload
-    def rename(
-        self,
-        index: Renamer | Hashable | None = ...,
-        *,
-        axis: Axis | None = ...,
-        copy: bool | lib.NoDefault = ...,
-        inplace: bool = ...,
-        level: Level | None = ...,
-        errors: IgnoreRaise = ...,
-    ) -> Series | None: ...
-
     def rename(
         self,
         index: Renamer | Hashable | None = None,
@@ -4734,8 +4722,9 @@ def rename(
 
         Returns
         -------
-        Series or None
-            Series with index labels or name altered or None if ``inplace=True``.
+        Series
+            A shallow copy with index labels or name altered, or the same object
+            if ``inplace=True`` and index is not a dict or callable else None.
 
         See Also
         --------