First question:
What actually counts as a necromancer for the purposes of versatile channeler?
Second question:
Does the Expulsionist qualify for the versatile channeler?
First question:
What actually counts as a necromancer for the purposes of versatile channeler?
A Wizard or Arcanist who has specialized in the necromancy arcane school.
Second question:
Does the Expulsionist qualify for the versatile channeler?
by the rules as written (RAW) no, expulsionist does not qualify since they are neither a necromancer nor a cleric that has the option to channel positive or negative energy.
If the inquisitor archetype could choose to channel negative energy I could see an exception being made, but they don't and so it doesn't even fall in line with what is intended to qualify for the feat.
So the only way to ever be a necromancer is to be a wizard or arcanist. That really seems so limited for the purposes of the versatile channeler feat. Basically saying you are either a cleric or a wizard/arcanist.