People talk about D&D editions generally with the idea that an edition reflects a version of the game with substantive rules differences. By that it is generally meant you couldn't simply port a character over, or an adventure without making changes that would alter a lot of how things work. Adding rules, such as new character class doesn't count as a new edition, changing core rules does.
When we look at the list of numbered editions people commonly use, we don't really see that however. Theoretically, 5th edition, for example should be, well, the 5th revision of the D&D core rules, but it plainly isn't. People have to talk about "zero" edition to refer to OD&D and Basic D&D isn't even in the equation.
So for fun, and without any of these being hills to die on, I took a stab at what the numbers should be, in my educated opinion, if we went in chronological order and numbered the rulebooks according to having substantive rule differences from previous releases. Basically, what I mean by substantive is that you can't play adventures from a given era using only a previous rulebook without running into major problems that would need to be addressed during play.
1) Alpha edition - 1974 OD&D
2) Beta edition - Supplements 1-4. Supplement 1 significantly changes core rules in OD&D, adds new classes, and creates a more complex game.
3) Gamma edition - Basic D&D (Holmes, B/X, BECMI, Rules Cyclopedia, Black Box)
The Holmes Bluebook rulebook largely adheres to "Supplement" D&D and it is tempting to lump them together. However Holmes notably introduces new movement and time rules that are carried into the rest of the Basic line. While all these editions of Basic continue to grow the rules base, the core rules are in substantial agreement throughout.
4) Delta edition - AD&D, 1st and 2nd editions. Many may balk that I have placed the entirety of 1st and 2nd edition under one heading, but the simple fact is that the core rules of both "versions" are nearly identical. Flavor issues and class tweaks aside the actual rule differences between 1st and 2nd are so minor they rarely even come up in play. Sure, new rulebooks came into the game throughout both these editions that kept adding things, and 2nd edition had loads of splatbooks and rule variants, such as in Ravenloft and Dark Sun, but those are all in the orbit of the core rules, without which none of the add-ons would function. These aren't separate editions in the sense of significant rule change.
5) Epsilon edition - 3.0 D&D
6) Zeta edition - 3.5 D&D. While 3.5 D&D billed itself as being only a revision, the truth is that the changes between 3.5 and 3.0 were quite substantial - far more than say between AD&D and OD&D + Supplements.
7) Eta edition - 4th edition D&D
8) Theta edition - 5th edition and D&D next. I've put these two together since "Next" was a free playtest that led to 5th but maybe one could argue otherwise?
Anyway, there you have it. I see 8 distinctive "editions" of D&D to date. Let me know what you think.