The limiting factor for the current carrying capacity of copper wire is the temperature rating of the insulation.
It's quite easy to determine how much power a length of copper wire dissipates at a given current. It's very difficult to estimate what temperature that is going to heat the wire to, as that depends on the surroundings, and how well they cool the conductor. The reason you have found multiple ratings is that some are for a single wire in free air, some are for bunched wires in a conduit, and some might assume different temperature ratings for the insulation.
As a very very rough rule of thumb, you can reckon on 10 A/mm2 for modest currents in single or paired wires that aren't deliberately thermally insulated, and 3 A/mm2 for the windings of transformers and solenoids of a modest (hold in your hand) sort of size. Needless to say, these figures come with wide margins of error, and are merely ballpark numbers for a 'smell test' of your initial designs.
Fortunately, the tempco of copper makes it very easy to measure the temperature of windings by measuring their resistance, it increases by about 10% for every 25°C temperature rise. Unless you have very reliable (expensive) thermal modelling facilities, it's best to just build something and see what your temperature rises are.
One further consideration, is this continuous operation, where it reaches equilibrium (isothermal), or short duty cycle, where the mass of the copper absorbs the heat and there's no practical cooling during the pulse (adiabatic). If the latter, you can calculate the temperature rise using the thermal capacity of copper. Or you can just calibrate it by delivering a pulse and then quickly measuring the temperature rise. Of course, the magnet still has to have time to cool down between pulses.
For any non-trivial electromagnet, cooling design is just as important as electrical and magnetic design.
