If I'm not mistaken, which I could well be in this case, it's a
time-varying electric field that "looks like" a current, and an
electric field whose time derivative is constant "looks like" a
constant current, at least with respect to the magnetic field to which
it gives rise.

To get an electric field with a constant time derivative, you need an
electric field that is forever slewing linearly. You need to keep
piling electrons onto one plate of the capacitor forever. So, you're
right that a constant conduction current can give rise to a constant
displacement current, but in any real capacitor with a breakdown
voltage, you can't get this to happen for very long, so it doesn't
have much applicability in any real system, and wouldn't get talked
about too much.

Dan