How to identify the C terminal on my HVAC
Your guess is correct

The C/3 terminal on your control board is indeed the C terminal you should hook your spare (blue) wire up to.  How do I know that? Well, your compressor contactor's coil needs to be hooked between Y and C, like a lightbulb connected from switched-hot to neutral with the thermostat from R to Y taking the place of the lightswitch going from always-hot to switched-hot, and we can identify the wire running off to the compressor from the Y terminal (its the other wire connected to the Y terminal, given that the thermostat is wired to it as well), so we know that the other wire coming from that compressor cable must land on the C terminal.