OP - if someone pulled out in front of me, giving me a scare - and then got out of their car to tell me off for my reaction, I would be really upset.
No, I would not yell and swear, because I’m not feral. But I would internally be so upset / angry.
Can you not see this from their perspective, as well?
They reacted badly - to something you did. Their reaction probably wasn’t appropriate, but people don’t always do the right thing when they’ve had a fright.
By telling them off, I’m afraid I think you escalated it.
The school have no place demanding apologies from anyone.
I think take a step back, get a little perspective, by trying to see it from both sides, and then move on and put it behind you.
It won’t happen again, because you won’t be pulling out in front them without looking again.