The pressure difference explanation is the basic explanation, though. The name of the force is "the aerodynamic force", and there's not really any confusion on that point.
The difference of shape between hydrofoils and airfoils is determined by the properties of the masses in which they move, explained by the same theories of fluid dynamics, rather than any fundamental difference.
That's just because when you angle the sheet, more molecules of air hit one side imparting part of their kinetic energy, and fewer molecules on the other side to counteract this. I do realize I'm explaining pressure on a molecular level here, but to me it's still "slicing through" and "pushing against" a lattice of molecules.
The difference of shape between hydrofoils and airfoils is determined by the properties of the masses in which they move, explained by the same theories of fluid dynamics, rather than any fundamental difference.