You’ve pretty much got it. If you get your own loan you just become a cash buyer, makes the actual car buying much simpler but maybe less wriggle room.
HP – dealers often talk in % interest but it might be flat rate, rough double that to turn it into APR (used to work for a finance company, people got quite upset when they realised they’d signed up for 20% APR!). Dealer gets commision on the finance when it’s above a certain %, they lose money if it comes too low.
PCP – you wouldn’t have paid off the car after the term. You’d either hand it back or pay a final lump to keep it.
You should easily be able to get a loan better than 9% APR at the moment