How the Current Flows in a Car?
When the battery is supplying power (discharging) to, e.g., the starter motor, the direction of the electric current is out of the positive terminal through the load and into the negative terminal.

Within the wire and frame, the electric current is due to electron current which is in the opposite direction of the electric current.

Within the (lead-acid) battery, the electric current is primarily due to proton (hydrogen ion) current which is in the same direction as the electric current.  

So, there are at least three currents to consider:  the abstract electric current (flow of electric charge), the electron current (flow of electrons, a carrier of negative electric charge), and the proton current (flow of protons, a carrier of positive electric charge).

Note that when the battery is charging, the electric current is into the positive terminal and out of the negative terminal.