AVI is an outdated file container that can contain any number of video codecs.
Fortunately I kept Quicktime Player 7.6.6 which continues to handle Windows-generated AVI files fine
That is because you have Flip4Mac installed. For others you need Perian. Quicktime has never natively supported the majority of AVI files.
To play such files you should continue to use Quicktime Player 7.6.6 with those plug-ins:
Flip4Mac WMV Player from
http://www.telestream.net/flip4mac-wmv/overview.htm (Windows Media Player for the Mac is no longer supported, even by Microsoft)
Perian from
http://perian.org/
You should read this support page
http://perian.org/#support in case you need to delete older codecs. In Macintosh HD/Library/Quicktime/ delete any files relating to DivX (Perian already has them). However it should be noted that Perian is not an internet plugin and will not play DivX files imbedded on a website. For that you will need the DivX Player browser plugin available from
http://www.divx.com/divx/mac/
