Although the title of the vehicle is in someone else's name, if you have documented proof you paid the balance of the loan for that vehicle, does it legally belong to you?

You can not go driving it without the title in your name.
You can take them to court and get an order to get the car in your name or get your money back.
At least that is how you have to do it in my state.
If you drive a car that is not titled in your name and the owner reports it stolen you will have a mess.