The starter isn't a particularly hard job-- no need to get under the car (I've done one outside in -10C weather), but unless you know that the one on your parts car has been replaced before, you are definitely on borrowed time. Mine died at about 150K miles. I agree with the others that you should either have it rebuilt or get a new one.

I went with a new aftermarket one because I needed it same day and I am in Canada where the selection is poorer and the prices are higher than the US. Previously I had replaced it with a used one, which died after about a year later. The failure on both of mine was a breakage of the copper pigtail between the solenoid and the starter motor. With this failure, there was no power to the starter motor at all, so it is different from yours, which sounds like a dying solenoid.

Good luck!

Correct, that problem should be the solenoid. While pulling the starter is not a lot of fun, using the donor car's starter for a rebuild makes a lot of sense.

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Keeping it running is better than buying new

When are you planning to junk the parts car?

If it is not soon, why not remove the parts car starter and have it rebuilt and put it in your driver. Then on a better day, put the suspicious starter into the parts car.

This cuts your labor a bit.