It does sound like the starter solenoid to me. I would check for voltage at the connector on the S/S when pushing the start button to make sure you don't have a problem with the neutral safety, or some other safety circuit. The S/S is on the right side next to the battery under the side cover. If you have your 11.6 volts there also check the fuse that is next to the connector on the starter solenoid. If all that check out short the two big terminals and the starter will crank then you will know it's the starter solenoid. You need more voltage charge or replace that battery.

 

Ahh. Now I had the same problem on my Pan European , and it was the starter button. There was a small amount of corrosion between the two contacts; a little rub down with emery paper, a dab of waxoyl, and hey presto.
The same can be done on the solinoid, although more often than not, it is a sealed item, if so , drill out the rivets.