Hey David,
depending on the max. current draw of your pump, you may connect it directly to the Relay output, or if it exceeds the RO1A specification on page -29/30- (PDF page 33/34), you need to contact an external relay that switches the pump on. The specification says: “Contact capacity: 3A / AC250V, 1A / DC30V”. This is a relatively high switching capacity, enough for a pump (but not enough for switching the dust collector or a workshop vacuum cleaner). Is it for the pump for the spindle coolant? If it is an AC 110–250 V pump which draws not more than 3 A current, you can connect the Relay output RO1A and RO1C directly between one wire of the pump and one “hot” contact of the AC wall socket, while the other wire of the pump goes directly to other contact of the AC wall socket directly. If it is a DC max. 30 V pump that draws not more than 1 A current, you need a power supply for providing the DC power for your pump and you wire it the same way as above, but instead of connecting it to the AC wall socket, you use the two wires from the DC power supply.
Regarding triggering and duration of the pump being activated, this is set with P0g.03 on page -79- (PDF page 84). You can set the condition to “spindle running” by setting P06.03 (Relay RO1 output selection) to “1” (In running) which would be the best choice in my opinion. This corresponds to your scenario Nr. 3. For the duration with which the spindle shall further be cooled after it was stopped, you can set P06.11 “Relay RO1 switch-off delay” to a value between 0 s and 50 s.
I took this information from Hitachi S1 Series VFD Manual (PDF).