Private party sales you are 100% correct. Taxes are based on where you are registering a vehicle correct.
Non-private party it is tricky and depends on the state you are purchasing from. IL has arrangments with about half the states to collect taxes on behalf.
So majority of the time I would collect taxes for an out of state purchase, and the amount would be based on that persons county.
This is incorrect. I just bought a 2011 Harley Ultra Limited in IL 2 months ago. I paid ZERO IL tax. I paid tax on $23,700 in MO 30 days later. It would work the same way for a car sale in IL from a MO resident.