or one side of jumper cable and just touch itIf you hot wire a big thick bit of wire to the 12v Battery+ and touch it on the starter solenoid 12v exit feed does it turn over immediately..
i suspect that delay circuit kicking in and out on load perhaps a loose or dirty load wire
i would touch the jumper to the starter then follow the wire back and touch it there
out of morbid curiosity i just turned off my ima batt, turned the key to run, waited until fuel pump stopped, turned the key to start, counted 1001 1002 and the starter engaged until release
sounds like yours goes GRUNT, then is commanded to stop trying, (i assume you are holding the key and never releasing) then it goes GRUNT and stops trying (or, are you turning the key to start over and over ) if you are HOLDING THE KEY to start and it is doing this, you need a meter on the selenoid wire to see if it is actually DROPPING OUT then re-energizing, this would mean the car is commanding that action and its in the brains.
if, on the other hand, you are turning the key back and forth, and it goes grunt grunt grunt start, i suspect bad connections that get hot from the interaction and expand and connect (which could also be in the starter OR the battery terminals or the battery itself, put jumpers on another car, dont start the other car, see what happens.).
Colorado? yep you need that starter to work before it hits single digits or less than zero, thats when mine kicks in once or twice a year on the first cold morning startup.