:-( it would appear that you need a bit of "refreshing" on how to use a meter. As roadie indicated, normally voltage measurements are taken with respect to a reference point... which in most cases is a ground... a known ground since voltage that you are looking for is a "potential" (energy) relationship from ground... ie. 12v is 12 "potential energy units" from ground. So when you took any of measurements and placed the "black" test lead of the meter at a ground point (ie. the - battery post... likely a ground but only IF the ground strapping and motor metal parts are in good shape), and placed the "red" test lead of the meter at a point, then those measurements likely mean something. Good.... but in your effort, the set of measurements that you did this for was when you put the black on the battery and the red at the "black wire" of the start... which just goes back to the battery... :-( basically all you did was test for voltage across your battery... OK... but not really of value since that could readily be done. It does confirm that the voltage remained there for the whole "key experience", which is kind of expected since the wiring is thick and unlikely to break....
In a round about way, you kind of did this for your other "technique" ... kind of in reverse by placing the black at the battery + (NOT the ground) and placing the red at the P wire. The problem with this "technique" is that you aren't really sure what is grounded... and since the + of the battery is a known potential above ground, any where you touch the black test lead may result in some form of circuit "connection" either directly or thru even some small form of "leakage" which can occur especially with windings of inductors / moisture / etc everywhere.
OK, enough... hopefully that will help in the future.
Back to testing, what you need to do is a good test like your other one with the black test lead at the -ve of the battery and the red test lead at the P wire... doing the same key things.... As others indicated, getting the key in start mode for this particular measurement would certain help a lot.
Lastly, when you did the "hot wire trick" how thick was the wire. It has to be at least as thick as the wire you see going in as the purple. Most people just use a screw driver...
If that doesn't cause the starter to make noise, you have likely found your issue... the starter, main motor or the solenoid.