Understanding fuse specs
Little fuse has this app note. It's very good and explains many of the terms. This one is also good.

TE connectivity has another good app note.

Your terms might be the same thing because it will cut off below hold, so perhaps there's confusion there. Usually the terms you're looking for is the difference between the trip current and the hold current, which is a hysteresis effect. The fuse will trigger at the trip current (lets say 0.5A) but once triggered, will stay triggered until the current drops to reache the hold or cut off current limit. This has obvious implications for the design to where you need to select those parameters to match what you expect will be short circuit or over current conditions.
In PTCs the hysteresis has to do with the thermal design. Once hot (which is the way they work), it requires the temperature to be much less than the temperature caused by the trip current in order to reset itself.