Yes you are correct, the drain pipe is the one that goes through the floor and to the back of the house.
At that point though is where the T is that the vent pipe ties into. This is right at floor level which will put the P trap above the vent pipe at that point. I was also wrong in calling it an S trap and stand corrected it will be a P trap.
The trap isn't actually installed as yet. I'm hoping the rest of my description makes sense.
Thanks