It plays two fairly different roles in MovieWriter (it's the stdin of
the long-standing ffmpeg-like subprocess) and FileMovieWriter (it's the
ever-changing handle of the frame currently being saved), with different
life-cycles and no reuse between the subclasses, so it's easier to just
inline it at the call site (and skip a comment about how
FileMovieWriter._frame_sink gets closed in grab_frame, hinting at the
tight-coupling between the two of them).

Also, no need to close the process stdin after calling communicate() --
communicate() does it for us.

PR Summary

PR Checklist

• Has pytest style unit tests (and pytest passes).
• Is Flake 8 compliant (run flake8 on changed files to check).
• New features are documented, with examples if plot related.
• Documentation is sphinx and numpydoc compliant (the docs should build without error).
• Conforms to Matplotlib style conventions (install flake8-docstrings and run flake8 --docstring-convention=all).
• New features have an entry in doc/users/next_whats_new/ (follow instructions in README.rst there).
• API changes documented in doc/api/next_api_changes/ (follow instructions in README.rst there).