I prefer the idea that this should be a documentation bug. IMHO it should be the element who directly holds parser output HTML from wikitext of a page.

No it shouldn't. $content has only ever been used by core and the extensions I could find to refer to the overall content area (including ui, etc...).

If you want the actual content text use $( '#mw-content-text ').

Updated the documentation.

Indeed. #bodyContent is never the right selection, it never was.

• The raw page output is #mw-content-text. This ID is fixed by the MediaWiki output handler and not overwritten by skins.
• The full page output is a few levels up, and depends on the skin. In Vector it is #content. Because this is variable, mw.util.$content helps.
• #bodyContent is somewhere in between.