Starter does not engage on manual car.

Replace the starter solenoid on the fire wall. If you jump it and still no turning. replace the starter..

Not enough information to answer. Need manufacturer, year, model, engine size, 2WD/4WD/AWD, and Manual or Automatic transmission as applicable in order to answer the question.

When you replaced starter did you notice the condition of the
flywheel ring gear? My guess is the ring gear is very badly worn or somehow the positioning of the starter is such that it is off enough
that it runs into the ring inside the gear. Did you hit something in the road before this symptom started? Any chance someone tried to jack up the car and used the starter as a lifting point?
Accident? Are you sure it's your original engine? Ie they used to be very easy to change. Any other thing that would bend the mounting point on the engine? If manual transmission, try to move the engine a little by moving the car with clutch engaged.
If Automatic try to move ring gear with a tool if you can get to it.
If this action is sufficient to move the ring gear into a "good" spot
and the Starter will engage, you now know you have a few bad spots.

Either the starter gear is slipping in the nose of the starter, preventing it from engaging or the starter is loose or has a stripped gear or you have bad teeth on the flywheel.After inspecting the flywheel, I would try a junkyard starter first.

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.

Sounds like starter solenoid is failing to kick in. The whirring is most likely the starter motor turning, but the teeth aren't engaging on the flywheel ring gear due to failed solenoid.

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This sounds like a possible starter problem, considering the age of the car. The solenoid switch may be defective or the bearings in the starter motor are worn out , causing the starter to not engage. Do a load test on the starter when you engage the starter switch. Require a test meter.

My guess is the selonid is not working providing the teeth are still on the starter gear.

The starter bendix is broken. Replace the starter.

Your starter drive is staying engaged. Either the starter has failed, or your ignition switch is keeping the starter engaged. Try moving the key back from "start" while the noise is occurring. More than likely, the starter is the culprit.