Sounds like there has been a massive cock up which is entirely the GPS fault, hence the letter. The threatening behaviour is completely unacceptable, and it's heartening that they've apologized.
I don't think you should leave it there though, as it leaves too many questions unanswered.
What did they think you had done that they would decide that it was appropriate to threaten you in a wholly unprofessional manner? And threatening to 'make life difficult for you' by someone in such a high position of power, power over your health and your university course, is a deeply unpleasant and threatening thing to do.
You need to know what they had decided you had done, as you need to be able to defend yourself against this, and also have it sorted out rather than have some accusation floating around the gp practice unresolved and likely to raise its head at any point in the future.
You need to know what processes or behaviour they have changed in order that this doesn't happen again.
I actually think you aren't reacting enough to the way this gp practice has cocked up repeatedly. You need to protect yourself a bit I think, by complaining and questioning what they are doing to ensure both fuck ups never happen again. You can do this in a nice way, you don't have to get angry or accusing, in fact it's better if you don't show those feelings, but they clearly need to change the standard of care they are providing you.
If your thought about someone on your course bitching and stirring is what happened, this is completely shocking and unacceptable. Listening to unsubstantiated gossip from another student and letting that somehow be treated as fact and the basis for the way they changed their treatment of you... that would be completely unprofessional and unacceptable.
If you can bring yourself to take this further, I think it would really help you. To make sure this never happens again, and to 'clear your name' and insist they clear your name in any notes or formal and informal perceptions of you. And to show that you need a much higher standard of care towards you, as they've seriously messed up twice this academic year... once could be a mistake, twice is very concerning and brings into question what the hell they are doing!
Sorry about the tone, I'm angry on your behalf!