Yikes, you are overthinking this enormously. First, I really think your life will be easier if you learn to disregard the arrows. As I have said, I feel they are nothing more than useless decoration to give students something to get wrong on tests.
The MOSFET, as used in integrated circuits, is normally a symmetric device. There is absolutely no difference between the source and the drain (don't call the source the "supply," as that's horribly misleading).
The only piece of information that you should gain from the arrows is whether a device is PMOS or NMOS. Of course, in CMOS circuits, this is very easy to tell just by their positions.
If you look at, for example, the NMOS device on the bottom of the inverter, you'll see that one of its terminals is connected to ground. The other is connected to the gate's output. This means that the terminal connected to ground is always at a lower potential than the other, and, I've said, that's the definition of an NMOS's source. For an NMOS, the source is always the terminal with lowest potential.
When the gate of the NMOS is more than a threshold voltage above it's source (ground), the NMOS conducts, and pulls the output to ground.
The opposite happens with the PMOS. Its source is the terminal with highest potential -- the one connected to VDD. The PMOS turns on when its gate is more than one threshold voltage below its source (VDD), and it pulls the output to VDD.
- Warren