Fused RED wire at starter solenoid leads to the "B" terminal of the ignition switch.
The other RED wire is leading to the terminal strip on the powerhead.
Is the battery in tip top condition and fully charged... at least 12.5 volts? If so, it's possible that you have either a loose cable/wire terminal end somewhere, a tight but dirty (corroded) terminal end somewhere, or a cable/wire that is failing internally. Any of the three would cause a voltage drop.
If you haven't already disconnected all cables and wires (one at a time) so that you could clean the cable/wire terminal ends and the components they attach to... do so to make sure that a voltage drop does not exist due to such problems. Then tighten all connections with either a wrench or pliers, not with ones fingers if wing nuts are encountered.
To test the solenoid and starter, and ground safety switch if said switch exists on the engine... using a known good fully charged battery, simply use jumper wires to apply 12v to the white wire on the 3/8" nut terminal of the solenoid. The solenoid should engage if the other black wire on the remaining 3/8" nut terminal is grounded.
Let us know what you find.