So first we have fuses and circuit breakers

And all kinds of fuses have defined characteristics like A,B,C ,..., fast, slow, lazy,...

Standards normally define a characteristic curve for fuses or circuit breakers. Usually these characteristics are very non-linear. This characteristic defines tripping points by multiples of nominal current. So while a minimum violation of the nominal current may require a fuse to trigger after minutes/hours, a violation by 3 or 5 might require the fuse to trigger in no time.

And while it's true, most fuses/circuit breakers work with a thermal principle, they are (always referencing the normal, common types) just monitoring the integral of I² over time.

https://en.wikipedia.org/wiki/Circuit_breaker#/media/File:Standard_Trip_Characteristic_of_a_Thermomagnetic_Circuit_Breaker.svg