yes, that is the solenoid for the starter.
The only thing you have proven is that the battery has enough juice to run the starter and that the starter will turn the engine.
Shorting around the solenoid does not prove that it is good or bad.

Apparently, when you hit the starter switch on the bike, it doesn't turn over.
To see if the solenoid is good or not, you need to put 12 volts on the control lead for the starter (where the Yellow with Black stripe wire is) and see if the bike turns over.
OR .. (preferred) just measure the voltage at that lead when you hit the starter switch.  If you get 12 volts, then the solenoid is bad.  If you don't, the problem is somewhere else.  (This assumes that the gound lead (black with white stripe) is properly grounded.  Measure that with an ohm meter)

Did you ever run a hot wire to the yellow with green stripe on the decomp timer? That ought make it spin over.
If that battery isnt charged up good, itll spin but wont start.

Everything seems to check out ok except for this device. Anyone know what this is?

Well we (we being my buddy) put a switch off an old gas trimmer to start the motor. Good enough, at least she is back on the road.