There is no rule or "official" definition. But in the industrial controls world, you won't see anything described as a "relay" if it is over 40A, yet you will see "contactors" that go as low as 9A.
As a gross general rule, the real difference is when switching inductive motor loads. With a 600V rated "relay", even if it is rated 35A for example, it's inductive motor rating is only at 240V max., and in this case, 5HP single phase, even if you have a 3 pole relay. With a "contactor" rated at 32A thermal, it can switch a 10HP 480V motor.
It's not really the current rating per se, its all about the arc quenching capabilities. Relays are not intended to quench a sustained arc that occurs when interrupting an induction motor circuit, where the back EMF from the motor attempts to keep the current flowing across the gap as the contacts separate. Contactors are SPECIFICALLY designed to do that, regardless of the current rating. That's why a 9A "contactor" will cost you more than a 20A "relay".