This is mostly true but one should add that
Bill developed as a short form of
William several centuries after Dick/Hick did for
Richard and Dob/Hob did for
Robert. We know this because of the existence of common surnames like Dix, Hicks, Dobbs, and Hobson based on those forms. If
Bill had been common back in medieval times we'd have a lot of people with surnames like Bills and Bilson, which we don't.
Bill only developed later, and most experts believe it probably started off as an Irish form that only later spread to England.