STARTER WILL NOT ENGAGE. - 1990 Geo Metro

Check by getting a screwdriver and bridging the terminals on starter if it doesn't turn try couole o wacks I know sounds funny but does actualy engage the pin

A lot of times and on most vehicles, when the Starter turns over but will not engage, that means the Bendix inside the starter is not pushing the starter gear out to engage in starting your engine. Best solution would be to just simply replace the starter. Sometimes you can have the starter rebuilt, but in most cases the fasted and simplest way is to replace it. Hope this helps

At the starter relay, you have two red or purple wires, one goes to the starter and the other comes from a 30amp fuseable link. Is that fuseable link circuit hot ?
From the key switch, power goes thru the alarm relay, then the clutch safety switch or automatic selector switch, then to the starter relay. The fourth wire on the starter relay is a ground.
You should be able to put power to the starter relay on the white or yellow wire coming from the safety switch to make the relay engage.
My guess is something wrong with the alarm system.

jumping across the solenoid will spin the starter but will not engage the pinion in the ring gear. Have a load test done on the battery as I suspect that there is a bad cell and you are not getting a full charge. If the battery is faulty then there is always enough power to turn on dash lights but nothing left to run the high current solenoid and starter.

You need to check for worn or broken gear teeth on the flywheel ring gear.

did you put right.

check your flywheel & starter, one or both will be missing teeth, or the starter isn't "throwing out" yo engage flywheel

sounds like you need to relace the starter

You can remove the starter and bench test it on a vise. You will need your battery or a charger to test the starter with. The trick to do it is simple. The grond is the vise and the positive is the battery cable lug to the starter. Have a small wire to jump the positive and simply touch the solenoid connector to see if the starter engages