Moving inside like that on a motorbike does a couple things.
1) It moves the center of mass of the system lower and to the inside of the turn. The bike does not need to lean as far over to stay balanced. You do not have to move the entire bike as far, which is a lot more inertia (plus the gyroscopic effects) and therefore slower than simply moving the person. The rider can shift over faster than the bike, so you are more nimble in and out of corners. You might do this with a bicycle if you are going through alternating turns rapidly - like slalom turns. This is not typical, so you don't see it. Plus a human powered bike is much lighter and speeds are much lower, so it's not so important.
2) It puts the rider in a position where it is easier to stay with the bike through continued acceleration. That is, the centrifugal force jams you into the side of the bike. If you sit on top, you have to squeeze with your legs or cling to the bars to stay attached. You need a seat to your side, not underneath you, in a high G turn.
Drawing lines through the bike tires to where the person is located and trying to compare to skiing is a bit of a stretch. The bike weighs a lot more than the person, so the dynamics and what is efficient changes. There could be a squid driving. Or nothing: