All the wiring looks ok.. A little rust on that power wire on the left. Pull that plug on the left just below the relay and check the fuse inside there...Inside the ebox is pretty dirty so Id clean that up with a paint brush and clean up the grounds on the bottom of the ignition coil..
12.5 seems a bit low.. Ive read, anything under 12.7 causes issues..
Pin 2-24 (RE-PU) is the wire that comes from that fuse plug in the ebox, but Id check it anyways and confirm its a 15 amp
the other RE-PU on pin 1-26 wire goes to the VTS so that seems good.
Pin 2-2 and pin 2-23 are the in/out from the relay to the start switch... We're getting there...
THe beeps with the lanyard are kinda hit or miss.. I have times where I plug in and get nothing... _BUT_ my info gauge lights up and I can crank.. Does yours light up when you plug in the correct lanyard?
As
@mikidymac said, the other lanyard shows the system is functioning by no allowing a diff lanyard to work..Id clean up the GSI lanyard with soap and water real good and try it again...
Im still leaning towards the relay being bad..Could also be the start switch...or low batt power.. All three are viable fixes in the troubleshooting tree.
If the relay is bad, it wont allow power to flow from the battery when the button is pushed..
If the switch is bad, it wont bridge the connection and actuate the relay...
If the batt is low, it wont do anything...