Case 1: your car is moving toward the west and the driver presses down on the gas pedal. You feel yourself being pressed back against the seat. During this instant, which direction is the car accelerating?

When we pressed again the seat from back side

So we can say initially we are moving with car at same speed

now when driver press the gas pedal we are pressed against the seat from backside

So it means the car moves ahead after pressing the pedal with larger speed and due to inertia we are still at our initial speed

Now the car will press us from back side as car want to go ahead with larger speed

so here we can say car is accelerating towards forward direction

so its direction of acceleration is in the direction of initial speed of the car