Firstly, may I paraphrase your question(s):
"Which preposition is suitable when mentioning an exam I took, how come the preposition is different in American to that of British English, and last but not least, do I need to use the definite article ('the') in either sentence?"
Final question first: Yes, you need 'the' if you want to indicate a specific exam (even in the understood context).
Why American English is different from English English?
I would argue that they like to express their differences (and that includes spelling and prepositions). But the logic is the real determiner: If Americans say 'on the weekend' it's most probably because they're in the habit of working on either Saturday or Sunday, so they only really count the weekend as one day. Then, it just stuck (even if some do have two days off.)
Which brings us to AT (one of your options). AT is used for events (on more than one day), places and time.
I would not use AT in your example unless you were only (doing something other than sitting the actual exam in that same location and at the time of the exam).
Another point is that we are GOOD or BAD at something, in general.
But we don't "do well" at something.
As for on or in, I'd personally go for on.
After all, there's an expression: to SIT an exam.