"Briton" is generally used as the noun form of "British", so when "Brit" is used as a noun - which is most of the time - it's abbreviating "Briton".
As for who gets to be called "Briton": In the loosest sense, anyone with residence in Britain can be counted as British when they're here, whether or not they're considered ethnically British (by themselves or others).
Bear in mind that "Briton" originally mean "an inhabitant of the British Isles before any of the Romans, or various flavours of Germanics turned up". There's been quite a bit of admixture since then. It makes sense - to the chagrin of the Welsh, no doubt - that the term has mutated a bit over the centuries.
Residence does not suffice: you're a Briton if you have British nationality (no matter of which of the 4 nations) so people have to apply for British nationality after at least a certain time as resident (5 years for Indians and a few other nationals, 10 for most foreigners) plus there are other mechanisms to get British nationality (for example, for descendants - resident or otherwise - of people born in Northern Ireland).
I lived for over a decade in Britain as an EU citizen but I'm not and never was a Briton since I never applied for British nationality.
Maybe not in any legal sense, no. How people and even news media use it, there's plenty of wiggle room.
e.g. allowing the ambiguity of "British home owner" to go unclarified, that is as "home owner who is British" as opposed to "owner of a home in Britain", and any similarly loose interpretations that go along with or derive from that.
I don't remember ever hearing "Brit" or "Brits" being used for immigrants in Britain and definitelly heard it used for when Britons are living abroad as immigrants (in fact when living in The Netherlands I had some colleagues who were in their own words "Brits")..
Always thought it was just another way of saying British and not going into the specifics of which nation in Britain did people come from (the whole "nations" thing for people abroad is generally irrelevant or even misunderstood).
I always thought of "Briton" in that last sense, while "Brit" has the meaning of anyone living in the UK (almost). But that's from an outsider's perspective.
As my English cousin corrected me, though, "I'm English, 'British' could be anything!". She wasn't, of course, talking about the difference between English and Welsh, or Scots.