Should I keep generic reference letters from my supervisors?
Let me answer the question in the title.


  Should I keep generic reference letters from my supervisors?


No.  You should not keep copies of reference letters, generic or otherwise.  Your supervisors should send whatever letters they write directly to the jobs to which you apply.  Those letters should never pass through your hands at all.

It's entirely up to your references what kind of letter to write.  You should always ask for strong letters individually tailored to each position you apply for, but there's no way to make anyone do that, or even to check whether they have.  Ultimately, you just have to trust the people you ask to do a good job, which implies that you should only ask people you trust to do a good job.