Pin_Guy is correct. I was going from memory and thinking it was a NPN. This is what happens when you get senile, Sorry about that.

The base (pin 1) will low when off and high when turned on.

The base (pin 2) is the lead that turns it on and off, high off low on.
The collector (pin 3) is the output the turns on the driver (TIP 102). A high on the collector means the transistor is conducting or a on state which will turn on the driver transistor. The emitter (pin1) will just have 5 volts on it all the time.
It is safe to test all 3 for the predriver, the collector of the TIP102 will have high voltage on it and should not be tested with a logic probe. It can be tested with your voltmeter.

You diode tested the predriver and it tested good but is most likely bad causing the driver to stay on. The best testing is done with the power on with the logic probe, in attract mode the base of Q-53 should be high and the collector should be low. If the base is high and the collector is high the then Q-53 is bad. If the predriver tests good then you may have gotten a bad TIP102, it happens.

It does sound like the predriver is working correctly. Since you have the coil disconnected you should be able to use your DMM and check the resistance of from ground to the collector (pin 2 or the metal tab) of the driver. In attract mode the resistance should be @ .900 meg ohms or higher. A low resistance will indicate a shorted Tip. You may have just gotten a bad one. Always good to test new parts first before installing, but still not 100 guarantee.

Sounds like a shorted TIP.

See sometimes they test good when they are indeed bad.

I still refer to my earlier post.

Yes you are testing correctly but different DMMs will give different readings so pick your favorite meter and get familiar with what it reads for a new TIP and predriver out of circuit. Check a good diode and note the readings too.

Swap all 3 parts just to humor an old man.

How about a pic of both sides.

I normally use a Xacto knife to remove the carbon deposits. If you need to build up the board use some super glue.

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You can buy circuit board trace repair kits that contain different size eyelets to make this repair.

You need to decide whether this is a concern depending on each situation. If it does need to be repaired then you can use a green colored nail polish for small repairs, you will need to mask off the solder eyelets first.

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What Pin_Guy is saying is, the best way to hold the solder pad in place would be to install a eyelet. He knows that the original didn't have one. Now this may not be very cost effective for you to do this for just 2 pads, that is why I suggested that you can just install a pad and solder both sides of the board as this will give the component far more holding strength then just soldering the top side of the board.

Check the ohms on the coil and compare to the chart. https://www.google.com/url?sa=t&source=web&rct=j&url=https://www.flippers.com/coil-resistance.html&ved=2ahUKEwjE5PeGwdjbAhXq6oMKHSVXCxkQFjAKegQIARAB&usg=AOvVaw06Jhg4YLUe59RxlJ1c1qp7

The coil resistance should be @ 22 ohms.