I think there are some nuances in thought as well here. I don't identify myself as a 'brit' - and I don't know of any British/Scottish/English/Welsh person who would. I personally don't like the term at all, but I understand that some people (primarily from the US) choose to describe us as such. I'm actually half English, half Welsh, so I would say I'm from the UK/British/a Briton.
There are some sets of countries that people (for whatever reason - primarily similarity of language & accent), mentally group together, i.e. USA & Canada and New Zealand and Australia. This means that someone outside those countries/continent may describe something as American, when they actually mean Canadian, or Australian, when they actually mean New Zealand. To someone inside those countries that might seem bizarre or offensive, but to an outsider, it's easily done, although no offence is intended. By using the term USian, it removes the mental ambiguity from American, which could mean Canadian or American (USA).
I think there was recently on another thread a quibble over someone saying something was 'typically European', when they actually mean German, Italian (or whatever, I couldn't work out what country they were talking about but it definitely wasn't the UK). As someone from the UK, that makes absolutely no sense at all - we don't identify as 'European', and our cultures and histories are so vastly different that it seems a bizarre statement. But from an outsider's perspective, we're lumped together geographically and (for the most part) economically, so what's the difference?
All in all, I think we should recognise that people outside our geographical area may not appreciate our own feelings about our identity, but as long as they're not disrespectful, we should try to take it in a positive light, and not see offence where none was intended.
This interested me, (as another Briton) I realised that which I would not normally self-identify as 'European', for the reasons you've given. An American friend of mine was in the UK recently. While she was planning her trip she made a comment in an email to me about how much she was looking forward to her trip to Europe, and my initial response was to think "that's funny, I didn't know you were going to Europe - I thought you were spending the whole week in England" (I did realise what she meant before I replied, however!)
That said, there are some situations where I would identify as both English and European.
(also, as the poster who inadvertently sparked this off by using 'USians, I've found the comments really interesting. I'd like to say that I had no idea that anyone would find it either unfamiliar or offensive - if I had, I would not have used it. I learned it from (American) posters on other boards where it's used simply as shorthand, with no negative implications at all)
I love pesto, but also use fresh basil in lots of other things. When I can get it. It may be easy to grow, but apparently does not survive being slept on by a large cat. (although it makes the cat smell nice for a while)