@SSethi (WMF): I adjusted the rewrite slightly to remove confusion over different "methods". There's only one method, it's just that it can play out in different ways depending how much the client assumes about which extensions are installed. On one end the client might assume everything and never use API:Authmanagerinfo (except maybe to get the captcha data), while on the other end the client might assume nothing at all and build its UI based entirely on the data returned by API:Authmanagerinfo.
The one remaining possible issue is the header that says "MediaWiki version: ≥ 1.21", but this page documents how it works after it was completely changed in 1.27. Trying to follow the instructions here on a wiki using 1.21–1.26 will quickly fail. API:Account creation/pre-1.27, which was moved to the bottom under "See also", documents how it worked for those versions; I wonder if that should be linked more prominently somehow.