What TheDirk suggests is a good, sound technique that will determine if the starter is being "told" to turn over the engine at every time you command it.
If you use your voltmeter, become familiar with the instruction manual for it. You will want to measure 12 Volts Direct Current or commonly labeled 20VDC with -- symbol. This sets the range of the meter to read dc voltages up to 20 volts.
Many voltmeters also will read VAC with the ~ symbol and ranges much higher, typically 250-750 VAC. This is for measuring household voltages and don't be playing with that. It is dangerous even while using a meter.
To measure current, commonly called Amps, the meter leads must be configured differently and you will burn out your meter, again if not done properly. Your meter's current or AMPS function is of little use here.
A small, insulated check wire from the small terminal of your starter up into the cab would allow you to verify your starter is being told to start when it finally fails on you. Assuming the large cable is well snugged onto the starter could be a mistake as well but usually, if a poor connection is the problem, it will not resume working until a new contact point is made by loosening the cable on the starter, giving it a 1/8 turn and torquing it back up.
Then it comes back to the starter itself again. There were different thicknesses and sizes of contacts available when rebuilding a starter. I think at the toyota dealership, there were 3 different sizes and the parts guy had to see what contacts were in mine to get the right ones. It is quite feasible that the rebuild shop used a thinner contactor connector and when the solenoid copper washer tries to make contact with the armature contacts within the starter motor, it sometimes don't make a good connection resulting in an intermittent operating starter motor.
Possible flat spot on that big copper plunger washer could cause the symptoms you describe and repeated attempts result in the washer turning as it was designed back onto a place where the washer can again make good contacts against the armature contactors.
Don't be taking all that dielectric grease out of the connector shown in photo above. A quick glance does not look to me to be melted. Plug it back in and confirm your starter is not getting its trigger voltage before you start messing with safety switches etc. If you are getting the trigger voltage, it is not the neutral/park safety switch. If not, then time to investigate why.