"Social Construction" and Equivocating on 'Race'

So, as I've made clear, I think that the main problem with the claim that race is socially constructed is that 'socially constructed' is disastrously confused/unclear. The phrase is used to mean an enormous number of radically different--in fact, mutually inconsistent--things. In fact, it's so polysemous, and used so indiscriminately, that it sidles right up against meaninglessness. To make matters worse, this polysemy is often used tactically in bait-and-switch operations in which one meaning is used for purposes of advertising some view as radical and Western-civilization-shaking, and then, when the view is criticized, advocates of the view retreat to a completely different, much more modest and easily-defended interpretation.
But let's ignore that problem for now.
Although I think this is oversimplified, I'll just throw a version of the claim out on the table for consideration: we can cast some light on the disagreement by characterizing it as semantic (and I do think that semantic elements are mixed up in it). Perhaps there are simply two different interpretations/conceptions of 'race' and race:
1. The races are white/caucasian, black/African, and Asian. Your race is a matter of your ancestry/biological heritage (and, perhaps, physical appearance).
2. Your race is perhaps partially a matter of your ancestry/biological heritage; but it's also partially (or perhaps exclusively) a matter of your "ethnicity," cultural heritage, language, etc.
Those are terrible, terrible definitions in terms of the details, but I don't care about the details right now. Sometimes it's the rough, general ideas that matter.
Anyway. Though we can't have a serious discussion that turns on the 'social construction' locution, we can easily avoid that by just saying things roughly like so: 'race' in the sense of 1 is a purely biological matter. 'Race' in the sense of 2 is at least partially constituted by non-biological components.
Now...when I grew up, 'race' clearly meant something like 1, and there was not the slightest hint of anything like 2. But 'race' is a vague term (which is ok, but can cause problems), and, apparently/perhaps an ambiguous one, and apparently a protean one, and so there's no reason to talk past each other on this matter.
I don't think this is exactly right...but it cuts a big swath down the center of the debate, and might get us thinking down the right path, even if it's not exactly true.