To be precise, the upward force the disk is the difference in the
pressure below the disk less the pressure above,

You're correct; equating pressure with force is a convenience to
simply discussions; it's a bit sloppy.

Force is defined as the time rate of change of momentum. Therefor
when the disk is neither accelerating nor changing mass there is
no force acting on it.

That's not a definition, it's an equation. *If* you have a change in
momentum, it's proportional to the NET forces acting on it.