A faulty coil with shorted turns will be less inductive than a properly working one, and if the overload is significant it will be mainly resistive. So it is the resistive load rating, not the inductive one, that matters when considering the impact of a faulty coil on the contacts controlling it.
Coils really won't last long at significant overload before self-destructing. A relay that is supposed to draw 0.1A at 24V but has so much shorted out that it draws 1A, will not operate because there won't be enough coil remaning intact to produce enough magnetic flux. If left energised it is likely to burn out completely in a few minutes. It probably couldn't take 5A without the winding immediately fusing open at the leadout, so will never present that amount of overload to the control circuit.
Contact ratings include an absolute maximum current carrying capacity that is governed by their resistance and the resulting temperature rise. This thermal limit applies to any type of load even if the contacts never switch it. Then there is a resistive load rating for one of a number of switching duties, which is a lifetime-imposed rating. The higher the current, the more wear occurs to the contacts at each switching cycle, so for a given duty and expected life, the resistive load limit might be lower than the thermal current-carrying limit. Then the inductive load limit is lower still, also depending on the specified duty category, because of the greater erosion of the contacts cause by the arc dissipating energy stored by the inductance. There is a certain trade-off between current and voltage, higher currents might be permissible at lower voltages because of the lower energy in the arc. Always subject to the thermal limit of course.