Starter will not engage

Sounds like your pinion/bendix gear is not reliably extending fully to engage the flywheel. There are three common causes of this: (most common) Dried up grease or dirt on the threads of the starter shaft and the pinion/bendix. Failing overrunning/one-way clutch in the bendix
replace starter.

Wrong size starter.
Incorrect tooth count on drive gear.
Solenoid not working ptoperly.

Can't be much, it's only a motor with a slide solenoid.

Wow. Probably the positive and negative connections have gotten reversed in the solinoid and starter motor. Still stuck? I recommend a DIYer's repair manual that shows electrical connections for the starting system.

You say your starter is kicking in..is it spinning and not turning the engine over? If so you have a problem with the flywheel teeth or starter gears not engaging..if no spin just a click..your battery is to weak to turn engine or you starter selonoid or starter needs replaced...good luck..

You will need a new starter rebuilt ect. to replace the one you currently have that is defective. you sound like you are spinning the starter motor but the starter bendix is no longer engaging the flexplate/flywheel on the engine.

The starter sprocket wheel is thrusted into engaging the flywheel by a big solenoid. That solenoid is integral part of the starter motor. If the starter turn itself but it doesn't turns the engine then either it was incorrectly installed or the thrust solenoid is defective.
Simple test: take out the starter, attach it to a bench, connect to it thew power cables from a battery (the minus one to the body, the plus one to the big connector bolt) then short the big connector bolt to the small one with a screwdriver - the solenoid should activate, the sprocket should be thrusted forward and the motor should start spinning it.
If that doesn't happen then the starter you just bought is defective, replace it. If that happens the starter was incorrectly installed on the engine. Reinstall it correctly.

Starter motor spins, but does not engage

1. Check the starter motor for a seized or binding pinion gear.
2. Remove the flywheel inspection plate and check for a damaged ring gear.

Does the starter kick out and spin normally out of the car? Have you checked the condition of the flywheel? Ususally the starter doesnt turn until the solenoid kicks the gear out to engage the flywheel. Check if the gear on the starter is bad, it should slip one way, and not the other.

check the battery if the starter tested good the starter spining and clicking and not engaging the starter suggest the solonoid is suspect it kicks the starter out might be going bad. BATTERY is the common factor with the three starters working about the same lenght of time