The difference between an Inrush rating and a HP rating on a set of contacts has to do with the PF when switching. Inrush in an AC load, TRUE inrush that can be as high as 20x the FLA, is only the current that has to establish the magnetic flux, so it lasts only a cycle or two at most, then drops off once inductance, and therefore impedance, is established. But for a motor load, the STARTING current will be 500-800% after that initial inrush, for as long as it takes for the motor to accelerate to about 80% speed, and because the PF during that time is very low, the contacts must supply all of that current. Worse yet is the possibility that the contacts should OPEN when the PF is very low like that, because the current is high, but because the load is inductive, the arc formed by opening the contacts is sustained longer and does more damage to the contact surface. So that high switching current at a low PF is then used to essentially reverse engineer the HP rating of the contacts.
Why the "inrush" rating on a relay contact without a HP rating? Transformers and anything else with a coil have inrush too, but only the extremely short duration magnetic inrush. The real root of the problem is the widespread misuse of the term "inrush" to mean "starting current" on AC motor loads. They are not the same thing.
Per the NEC and I believe UL508A by the way, you must have a HP rating shown on anything that is switching an AC motor load, you cannot "roll your own". There are specific tests done for attaining those HP ratings. That's why often times people are confused as to why the HP rating of a relay or contactor is seemingly so much lower than the "thermal" current rating, it's all about the switching capability, not just the current carrying capability.