After charging it was 12.9 on the multimeter.
Good point and helpful hints there ...
The quote above makes me hope he has got a multimeter by now. Remote diagnostics are always a difficult thing and especially your proposal regarding the voltage drop on the starter's main power cable is very valuable.
Zanzibar
The voltage drop would show us at a glance whether the cables and ground connections are healthy!
In the past I've frequently seen excessive voltage drop due to dirty, corroded cables / cable clamps or ground connections (bad connections sometimes get extremely hot when cranking the engine), worn out starter motor brushes and some more issues.
At least the ignition switch and starter solenoid seem to be ok because the engine starts cranking instantly when turning the key ...