Copper wire current ratings
Copper melts (the wire fuses) at 1984 F (1085 C). To avoid a fire in your house or equipment, you will generally want to keep the temperature of wires very much below that, so you'll need to operate them with currents far below the fusing current. 

Ampacity tables are based on some particular choice of the maximum temperature rise of the wire (as well as assumptions about airflow around the wire, insulation thickness, etc). We usually limit the temperature rise to something like 25 C to avoid temperatures that could disturb the operation of other parts of our circuit, degrade the wire's insulation, etc.

As you can see, this limit is very, very far below the fusing temperature of the wire.


  Moreover, which of these estimates is most useful for estimating the maximum current that is safe for use in an electromagnet (both with and without insulation)?


Hard to say. On the one hand packing the wire close together will tend to reduce the ampacity. On the other hand the core of your electromagnet will provide some conductive thermal dissipation (heatsinking). 

You could use a FEM solver to find the thermal limit, or just build one and see how hot it gets with different currents through it. After you have built 4 or 5 you will probably have a good idea what wire gauge you need to support different current requirements with a given geometry.