Is anti-seize paste 'just' copper grease?

This one might well be. Presumably made for brakes. What I would call "proper" anti-seize paste will normally contain molybdenum disulphide and, IMHO, is normally the thing to use in other domestic applications.

Expensive, but a little goes a long way. (This one also contains graphite).

In the power industry, you have to be careful with Moly Disulphide in high temperature applications, typically involving stainless or high strength (creep resisting) steels, also in the presence of water at elevated temperatures (as in feedwater and steam systems). Here, pastes containing finely divided nickel are sometimes used.

This one is also good

I've used Copper Slip on the nuts and bolts on steam turbine engines in ships engine rooms.

I've always put a dob of Copperease on the screws that hold the glass in my wood stove. Time will tell if it helps :)