Starter won't engage when key is turned.

AS above, here's how these work.

The coil of the relay is hooked to the two "push on" terminals, which are interchangeable. One is hot "in start" coming directly from the IGN switch, and in order for the relay to pull in, the remaining "push on" terminal must see a ground from the neutral safety switch.

Yes, the "s" wire (traditionally yellow) should be "hot" in "start."

I would first hold the key to start, then wiggle the shifter in/ out of park and neutral. If you get a start, or kick out of the starter, you've found the area of the problem. NSS could be bad, or linkage out of adjustment.

A further check would be to use a clip lead, and ground the relay terminal running down to the neutral safety switch. This should result in the engine cranking in "start" CAREFUL it will now crank "in any gear."

If you don't get a crank or relay "click" with the NSS terminal grounded, now try pulling the yellow "start wire" and jumpering from that exposed "push on" relay terminal over to the big battery stud terminal. If this final act results in no crank, replace the relay.

Also, if you can hear the relay click but the starter won't, jumper from the big battery stud to the "square" solenoid terminal. If the starter operates, and you can hear the relay click "with the key," you also have a bad relay.