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I believe you are correct; OD&D is the holmes/moldvay/etc. books that existed before AD&D, but I'm not certain.

Nope, that is Basic/Expert, and eventually BECMI/Rules Cyclopedia. These were released in parallel with AD&D.

In theory they catered to different playstyles, in practice it was largely to escape paying Arneson royalties.

OD&D is the original 1974 release co-authored by Gygax and Arneson

Fascinating, thanks! One of the things I love about D&D is that there's so much historical density to it.