Why use a current limiting breaker?
Could those breakers be applied to higher short-circuit current than they are rated for?
What is the range of current those breakers should be used and why?
Why some manufacturers offer them while others don't, or maybe they just called it differently?
A current limiting breaker exhibits a current limiting characteristic. That is it can reduce the peak apparent fault current to a lesser value and isolate the fault in less than a half cycle time. They cannot interrupt currents higher than their nameplate short circuit rating. Breaker frame ratings of 150A and 250A tend to have the best current limiting characteristics. Some 600A frame breakers can be current limiting, but not as current limiting as the smaller frames. Most breakers rated 800A and higher are not current limiting. Some current limiting breakers were developed to obtain higher interrupting ratings than previously available in older non-current limiting designs. The current limiting threshold is typically greater than 10kA.
Current limiting circuit breakers are used to protect loads close to supply source (substation). Other application is cascading in order to get economic benefit of current limiting capability of breaker in installation. Fuses are also current limiter and cheaper. Current limiting circuit breakers ensure less thermal stress on cables during short circuits.
There are some manufacturers that have a specific blow open contact type that produces a dynamic impedance that is supposed to be current limiting without using fused limiters, but I don't have any direct experience with these.