DOC: Move Animation and MovieWriter inheritance diagrams to their sections

Inheritance diagrams belong next to the text that discuss the shown
classes.

Also:
- remove `:private-bases:` option. - We don't have any private bases
  here so it did not have any effect.
- exclude `ABC` as the top class for writers by specifying
  `:top-classes:`. It's irrelevant that we inherit from `ABC`.
- add the missing `PillowWriter` and `HTMLWriter`

Author: timhoffm

doc/api/animation_api.rst

@@ -12,23 +12,15 @@
    :backlinks: entry
 
 
-Inheritance Diagrams
-====================
-
-.. inheritance-diagram:: matplotlib.animation.FuncAnimation matplotlib.animation.ArtistAnimation
-   :private-bases:
-   :parts: 1
-
-.. inheritance-diagram:: matplotlib.animation.FFMpegFileWriter matplotlib.animation.FFMpegWriter matplotlib.animation.ImageMagickFileWriter matplotlib.animation.ImageMagickWriter
-   :private-bases:
-   :parts: 1
-
 Animation
 =========
 
 The easiest way to make a live animation in Matplotlib is to use one of the
 `Animation` classes.
 
+.. inheritance-diagram:: matplotlib.animation.FuncAnimation matplotlib.animation.ArtistAnimation
+   :parts: 1
+
 .. autosummary::
    :toctree: _as_gen
    :nosignatures:
@@ -170,6 +162,10 @@ Examples
 Writer Classes
 ==============
 
+.. inheritance-diagram:: matplotlib.animation.FFMpegFileWriter matplotlib.animation.FFMpegWriter matplotlib.animation.ImageMagickFileWriter matplotlib.animation.ImageMagickWriter matplotlib.animation.PillowWriter matplotlib.animation.HTMLWriter
+   :top-classes: matplotlib.animation.AbstractMovieWriter
+   :parts: 1
+
 The provided writers fall into a few broad categories.
 
 The Pillow writer relies on the Pillow library to write the animation, keeping