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.
As background, I work for a US Federal government agency. I am physically located at an agency center, but hold courtesy adjunct appointments at two universities where I serve on graduate committees, mentor undergraduates, collaborate with faculty, and attend seminars. Additionally, my agency has cooperative research units (or coops for short) located at universities.
We have a fair number of cross-overs where we hire faculty to join us as researchers or have our researchers leave to join university faculty. This is true both at the post-doc level and more advanced levels. Personally, when I started as a post doc with my agency, I did not know if I would become permanent staff (I was lucky and did, in my case).
Answer to your title question: I consider myself part of of a broader academic community, but more tangentially involved rather than an academic. So, no I am not an academic. However, my colleagues who are part of the coop units, generally consider themselves to be academics because they are fully embedded in universities and must hold academic professor positions as part of their jobs (and they get both *.gov and *.edu email addresses).
Answer to your underlying question about how to market yourself: I am a research scientist. Many parts of my job are similar to a research-focused academic (e.g., I plan and conduct studies, I apply for funding, I publish, etc). I would sell yourself as a researcher. That's the role you want fill. For example, at a research institute, you would be conducting research. Or, at a university, you want them to hire you to conduct research. Last and pragmatically, if you're not able to get in with a research institute or university, industry hires researchers, but industry almost never hires academics.