All superconductors superconduct, but there's wide variation in how strong of a magnetic field they can endure before leaving the superconducting state: https://www.researchgate.net/figure/Upper-critical-fields-an...

ITER (and the LHC) used NbSn magnets, since it was designed in the 90s. Newer designs will use newer superconductors.

Think of it as the amount of energy you can store in a superconductor.

This is talked about in terms of the ‘phase diagram for superconductors’

http://www.supraconductivite.fr/en/index.php?p=supra-levitat...

As you put more current in the supercondutor, eventually you break it and it leaves the mode, as it gets hotter, likewise, as the magnetic field gets stronger, likewise.

You want the largest ‘area under the curve(s)’ to handle more and more ‘superconductivity’

New materials are enabling this which has the cubic effect mentioned for reducing confinement volume and making things more affordable and increasing energy densities that can be contained, etc.

High temperature superconductors provide more stable resistance-free conduction than traditional superconductors. If both are cooled to liquid helium temperatures, that increased stability translates to being able to sustain a higher magnetic field. Higher magnetic fields provide tighter plasma confinement and higher density, which allows for much smaller fusion reactors.

They still saturate. High-temperature is nice, but a separate consideration.

HTS coils would still be run with liquid helium for higher critical current and higher magnetic field.