Except the people I talked to at the time actually did seem unaware of it. They were people who'd come into the hobby after OD&D was a dead letter, and simply didn't know it existed. They knew about B/X and AD&D but were not aware anything predated the former (in fact, in at least one case, they were really puzzled that AD&D had gone "away" from race-as-class).
Apparently the people I talked to either hadn't noticed or didn't know what "white box" referred to.
I think there's at least a big enough difference between some of them mechanically (as noted, the whole race-as-class thing seemed positively weird to me when I heard about it, and certainly far more inflexible than needed (I already thought the lack of PC dwarf clerics was pretty odd) that the distinction between OD&D proper and the offshoots is not trivial.
This pretty much reflects my experience of it.
I started with D&D (that's what we called it at the time and was the default of what I called it until...well...a LONG time later). It's now known as OD&D, but I just called it D&D. That was the D&D for many years.
When Holmes came out that was more of the New box introduction. I don't think I actually had a name for it. I was playing D&D with the guys but never had a set of rules until I got the Holmes box. Later I got my own set of what we now know is OD&D. I intermixed Holmes and OD&D when I finally got my set. Didn't get AD&D until later.
I started AD&D and played that in place of OD&D (which I knew as just D&D).
I just knew BX and BECMI as Basic D&D. I got both of them mixed up and jumbled together for many years. Even now, if I want to play a game of BX or BECMI I'll combine rules from both sets together and have them intermixed. Normally I think I prefer Mentzer's Red box but use BX's Expert box in play. It is probably because I prefer the variable damage and such to other items.
Even with all that, I still kind of called it the Basic D&D or Basic set (of course it had the rest of the sets, but didn't start calling it BECMI or BX until after 3e came out if I recall correctly, as others started these naming conventions).
I've found over the years that VERY FEW were actually familiar with what OD&D was actually like at the time. S&W is a good example(of what it appears as at least to me) of someone who took a good guess at what it was like, but was more familiar with BX than with the white box (at least it APPEARS that way). I pointed out some of the items from the document when I d/l'ed it. Beyond just the elf, they had variable weapon damage, Armor class below 2, To hit tables below 2 AC, etc...etc...etc. That's the white box version they originally released...not even the core rules. Core rules it gets even MORE BX centered from the original rules.
It's not a big problem, not a bad thing, but I tried to point out what the original Three booklets were like from long ago to those who thought it was basically BX or BECMI. If they CONTINUED to make claims it was OD&D after being corrected, that was a little irksome at times.
PS: PERSONALLY, though, these days, I refuse to play OD&D UNLESS we are using the Greyhawk supplement. Greyhawk was SUCH a GAME CHANGER in the game. It is like night and day of a difference between playing with it and not playing with it.
If something is completely based on the 3 LBBs though and claim it, they should be faithful to the 3 LBBs. On the otherhand, including all of OD&D or some supplements is more of my preferred style, but I don't know what OSR products actually claim to be based on the complete or all of OD&D as a whole (or the 3 LBBs+supplements or some of the supplements, etc).
Would be fun to play an OD&D based OSR game based on the 3LBBs + some of (if not all of) the supplements. Toss in some of the strategic review stuff and it could be really interesting. (in the change to AD&D some of the stuff changed from the supplements and Magazines. Some of the classes were different for example).