The tire generates forces... magically!!!
No, there is a force acting on the tire and reacting on the tarmac. Newton said that and I believe him, most of the times.
Now, you can decompose the force acting on the tire, anyway you want. Usually is fine to use 3 (vertical, longitudinal and lateral)
"You need great passion, because everything you do with great pleasure, you do well." -Juan Manuel Fangio
"I have no idols. I admire work, dedication and competence." -Ayrton Senna