Why is a MOSFET triggered by Vgs and not Vgd?
The symmetrical cross-section as it's usually drawn doesn't quite agree with the actual structure, which is highly asymmetrical. Actually it looks more like this:  

  

with a much larger area for the drain than for the source. It's possible to specify \$I_D\$ vs \$V_{GD}\$, but you'll get a totally different relationship, which is rather irrelevant in common source applications, which are the most used.