A Variac delivers a variable AC voltage. The smaller the wire, the easier it is for some amount of current to melt the wire. A fuse is basically wire that melts within an enclosure (glass envelope) so it doesn't cause a fire.
Direct Current or DC is better suited for a magnet. The amount of magnitism is related to the current through the wire, not voltage. As the wire heats up, if a fixed voltage were used, the current would be less. Large commercial magnets like I worked with had a magnetic field of 30 kill-Gaus and was water cooled.
Because the magnet is composed of a low reistance wire, the voltage required is low. The magnet I used required about 100A of current at a low voltage, under 10 volts or so.
DC power supplies can be unregulated, constant voltage, constant voltage and combinations such as constant voltage, current limit. Voltage, current and power can be limits where the power supply can operate. In order to have power limiting, the power supply would be called an Electronic Load. I've barely scratched the surface.
Your car has a battery with a short circuit current of 400 Amps or so (Cold cranking Amps) and each circuit is protected by a fuse, so it's all or nothing.
You really need a constant current power supply with a low compliance voltage. For critical situations, water flow and magnet temperature are used as interlocks.