A Mass Hanging from a Spring

This. Somewhat.

Assuming zero energy loss,

Reduction in Potential Energy = Gain in Kinetic Energy

So not all the Potential Energy gets converted. If the pendulum bob is still a certain height h above the ground at the bottom of its swing, it will still contain some Gravitational Potential Energy. Still, delta P.E. = delta K.E.

Simple as that.

mgh potential energy is based on a referential height. You could always say it's from the lowest part of the swing. If you want REAL potential energy then you should go with this one that accounts for how the gravitation field varies with your distance to the center of the earth:
V® = -GMm/(Rt + r) where r is your distance to surface of the Earth and Rt is the radius of the Earth.