New Farmer Looking for Advice - LT180 Won't Start

Hi guys - I just bought a small farm and would really like to start learning more to be more self sufficient to make my own mower repairs instead of sending it into the shop. Before yesterday I didn't own a multi meter and had no idea how to do any of this - I've just been watching lots of youtube videos.

I have an early to mid 90's John Deere LT180 that ran fine last week, but now won't start. When I turn the key to start, I get one clicking noise - then nothing. When I put my finger on the starter solenoid I can feel it vibrate - so therefore it seems that is making the clicking noise (which I don't know if that is normal or not). The engine never turns over. Headlights work fine.

Here is what I have done so far:
- tested the battery with my multi-meter, and shows 12.5 volts
- tested the 2 pole starter solenoid. registers 12.5 volts at first pole, and 12.5 volts on the second pole when I turn the key.
- tested the seat switch, the break switch, the PTO switch and the ROI switch all for continuity and reverse operation when triggered.
- tested the purple wire at the solenoid. 0 volts when key is off. 12.3 volts when key is turned to start.
- tested the ignition switch
--- when in off position, continuity between ground (G) and magneto(M).
--- when in run/lights position, continuity between battery(B) and A1, and lights (L) and A2.
--- when turn the key all the way to start, continuity between solenoid(s) and battery(b)

What else can I try? How do I know if I have a bad starter?

Any help is appreciated. Thanks!