OD&D>AD&D and OD&D>BD&D were related. There's a 1977 Dragon article that explains it all. Essentially, what was originally a supplement to a supplement of a war game had grown into a pretty popular game in its own right with rules scattered between a boxed set, two sets of miniature rules, four more supplements, and a bunch of The Strategic Review and The Dragon articles. Plus there was a bunch of other stuff out there that was quasi-official that a lot of fans didn't know what to do with.
Thus TSR decided to do 1) a comprehensive 4 volume version of the game for the dedicated fans (Advanced D&D) and 2) an introductory boxed set for beginners (Basic D&D). OD&D was to be continued for the fans of that version of the game, and printings of the boxed set and the Supplements continued right up to 1980.
The reasoning for BD&D>B/X D&D seems to have been that BD&D sold beyond TSR's expectations (per a 1979 Dragon article) and they wanted a more coherant progression from the Basic box. The Basic box didn't exactly flow into either the AD&D or OD&D rules. So, they made the new Basic>Expert boxes, which were still essentially the OD&D rules minus all the supplements, for beginners and those who didn't want a rule-set as heavy as AD&D.
The only reference from anybody regarding Dave Arneson in the D&D re-designs that I've heard was from Frank Mentzer in a post on Dragonsfoot with regard to the design of his edition (1983+) of D&D.
AD&D1>AD&D2: Gary Gygax announced that he and Frank Mentzer were working on a 2nd edition in 1985. This was largely going to be a consolodation of existing rules, plus the addition of a few more classes, because, as with OD&D, the rules had begun to be scattered over a plethora of rule books, magazine articles, etc. It wasn't going to be the radical re-design that actually occurred. Work had progress on it through 1987, when both Gary and Frank left (were booted from) the company. Dave 'Zeb' Cook and Steve Winters began 2nd edition all over again. They're design goals seemed to have been:
1. Forstall any IP rights that previous authors may have;
2. Pacify the enraged mommies;
3. Make the game as customizable and generic as possible, while holding onto the same basic design.
Of course, they concentrated more on the second half of that clause in #3, whereas the 3e designers concentrated more on the first half of the clause. The cynic in me says that #3 was done simply to sell more rules supplements, although I'm sure a lot of gamers were arguing TSR was giving the fans what they wanted.
R.A.
(I want to not get involved in any flamewar, 
)
