Please post engine model number and code. We can look at schematic and walk you through appropriate testing procedures.
Put your multimeter on continuity, looks like a little speaker symbol. Touch probes together and listen for meter to make beeping sound.
Next, put red probe on metal surface of solenoid and black probe on negative (black) terminal of battery. Let us know if the meter beeps. This will let you know if you have proper ground.
Next, change volt meter to voltage VDC, same as you used to obtain battery voltage of 12.6 you posted earlier.
Disconnect wire from spade on solenoid, place red probe in female end of wire. Place black probe on battery ground. Turn key to start, ensure all safety mechanisms are met. Ie .....parking brake on, seat down, pto off etc.
Provide voltage reading sir.