Surely they should occur at completely different times?
Minimum fuel = we can absorb the delay that as stated, but we will burn alternate fuel to do so; emergency = the goal posts have moved, and we are unable to absorb the new delay or divert without using some final reserve fuel.
Regerding fully serviceable aircraft, I can't think of a scenario where one would have to declare an emergency without having made a minimum fuel call some time previous. Is there one?