Yes there most certainly is a way. It's called your driving record, and it's the printout you got. That's how many active points are on your record at the moment.

You know what that number is, because you've got a copy right in front of you. That's exactly what anyone, your insurer, employer or a judge will see when they look at your driving record. Your state / provincial driving record is the first, final and only authority on your licence and conviction history. If something is on there that should not be there, you need to clear it up with the licensing authority where you live. Because nobody else, no exceptions, will take your word over a government record.

News flash: If your insurance rates go up because of a not-at-fault accident, they would have gone up that much anyway even if no accident had happened. Yes they would.

When your premiums go up even though you've had no at fault accidents and no tickets, it's because of inflation. Auto insurance is a for-profit business, not a public service. And businesses, all of them, are subject to inflation. Inflation is a very real thing, and it affects insurance companies and everyone else on the planet. In recent years, some insurers around the world have increased their prices (for everybody) by 25% each year. That's a lot.

But to answer your question, yes a private settlement is totally legal. If I do $100 or $10000 worth of damage to your vehicle and pay for it out of my own pocket, it's legal and legitimate.

If you are in the U.S. you can technically argue against any tax on your vehicle before the date that the law as enacted [(ex post facto) after the fact]. I would say contact your local DMV or the secretary of states office.