* Where you keep your car
* Typical mileage
* Car model & age
That's probably it. In addition they shouldn't be allowed to use age as a discriminator (we don't do that for National Insurance in the UK and it works well).
It's ridiculous that they ask for things like your job and your marital status.
Why is it okay to ask for the car model? Yeah you can say that very old cars or specifically unsafe cars might have worse risk profiles but most modern cars (made in the last 10 years, for example) are more or less all road safe yet their risk buckets are different. That's not because of the car itself, it's because of the kinds of drivers who end up buying them. So we are back to the 'hotmail' argument.
For third party actually I think you are right - it shouldn't depend on the specific model you drive, but it could depend on the power, e.g. you're more likely to crash in a fast car.
The point is insurance premiums shouldn't depend on things that you can't realistically change like your age, medical history, salary, marital status etc.