Do people who work at research institutes consider themselves "academics"?
I don't have statistics, but I'd guess that if you asked the question of individuals working in such institutes it would depend on exactly how you asked the question. I would think that the quick answer would be no, but a more reflective answer, when provided with some "definition" of academic might be yes, but would depend on that definition. 

I'm also going to guess that only a small proportion of members here are from such institutes but not associated with a university. 

Of course, if a person moves from an institute to a university they would then likely self-describe as an academic. 

But if you just ask "what are you", you'd be more likely to hear "researcher" than "academic". 

But, as you say, it is mostly semantic, given that many academics do just about the same sort of thing as institute based researchers. 

The situation is complicated a bit, of course, since many such institutes are associated with universities - some very closely. It might be further complicated in situations in which industry researchers have some "training" duties, even serving as committee members for graduate students.