Electronic – Electric potential and the flow of electrons in a battery

Does electric potential influence the direction of current?

Yes it does. As you said, conventional current flow is from a higher potential to a lower one.

With your typical AC waveform (a sinusoid), the voltages don't switch discretely. They gradually increase or decrease. For one half of a cycle, one terminal will be higher than the other, and you will have current flow from the more positive to the more negative. For the next half of the cycle, the other terminal will be higher, and you will have current flow again from whichever terminal has a higher potential relative to the other. This is conventional current flow.

Electrons are flowing from negative to positive when there is a connection for them to flow. You can also think of it as "holes" (or the absence of electrons in an atom) flowing from positive to negative.

Because of historic reasons, we generally think of conventional current, or the "holes", rather than the actual electron current. You are correct that the positive terminal just "pulls negatively charged electrons."

There is also no magical node called "ground." We simply assign a point that is easy to keep track of to refer to as ground. In a circuit, you could refer to the positive terminal of the battery as ground, and simply be working with negative voltages (though it would likely cause some confusion for others interpreting your work).

Likely in the case of shorting the battery to the chassis of the car, the negative terminal of the battery is already connected to the chassis. Possibly this is done for some sort of EM shielding (vehicles are inherently very noisy environments). When you make another connection from the negative terminal of the battery to the chassis, no current flows. When you connect the positive terminal, current flows through the chassis of the vehicle to the negative terminal.