I don't buy the premise. Yes, people who live abroad as part of an organization sometimes have access to infrastructure the average person abroad does not have, but that doesn't mean they are insulated from contact with the culture.
I've known enough of both missionaries and military folk to know that their support system is less substantial than you may think it is (especially the missionaries, who are often utterly alone), and their contact with the culture is deeper than you might expect.
Moreover, no matter how strong their existing support system is, they may still have issues that are best explored outside that system. Consider the case of a woman who is on birth control not primarily for contraception but to regulate her hormonal cycle. Whether military or missionary, in either case it may be embarrassing or even professionally detrimental for her to ask questions within her organization about sourcing birth control in the new country. Colleagues gossip, rumors spread.
Are we supposed to turn her away saying "No, I'm sorry, your experience is too defined by your organizational role. You're not expat enough for us."?
I worry that this site has a risk of becoming, to coin a word, expatronizing.
I understand that it is important in the early days to try to define clearly what the site is and is not about, but I think there is a very simple guiding principle for that, which draws on the premise underlying every StackExchange site: Will this question connect the questioner with an expert who can help them, and the expert with a question they feel inspired to answer?
There are cases where someone will come to this site and ask a question that no one has the expertise to answer, such as "What are the medical facilities like at the Rammstein Air Base?" And in those cases, we will tell them we don't have the expertise, and a military families' forum may have the information they need. But if they're asking "Will I be able to buy a car in Japan without relicensing myself under a Japanese driver's license?" this is the place to find that expertise, regardless of what organization is ostensibly structuring and "defining" their experience.
This is very similar to the situation StackOverflow faced with homework questions. There was debate for some time over whether those questions should be answered, how they should be tagged, etc. In the end, the homework tag was deprecated because the source of the question is not as relevant as its content. If the question offers a legitimate problem to a community of experts who have legitimate expertise to solve it, then it's a good question whether the asker is a student or a professional.
Sort-of-but-not-exactly expats are going to be the "homework questions" of this site. I hope we will choose to answer them on the basis of the quality of the question itself, not the specifics of the asker.
p.s. Full disclosure: I am definitely a "homework question" variety of expat myself. I've bought groceries hundreds of times while working and living in a place, but I've never established a permanent new home. I'll be on this site as more of a questioner than an expert most of the time, so I certainly have a bias of personal investment in the arguments I make above. The expat lifestyle I aspire to is more of a "live one or two years in one place, then move to an another, freelancing along the way" one that lacks the kind of permanent intention some of you seem to see as essential to being an expat. I'd like to feel welcome here, and I'd like to see students and missionaries and oil company contractors and military people welcome here, too.