The object of the exercise is to bypass things one at a time till you verify if one of the parts is duff.
So when I said to jump directly from your battery, I meant to jump both sides, the power & the ground
If the battery can not spin the starter then either the starter or the battery is duff.
So the next cheapest & easiest thing is to bypass the battery.
If you other battery spins the starter then you battery is obviously the problem.
If jumping the + side from the mowers battery to the starter spins it then either the solenoid or the cable between the solenoid & starter is duff.
So you would jump that cable and try the key to see if it was the solenoid or the cable.
If you get nothing jumping the + cable then the next trick is to jump the - side of the battery.
Engine cranks then you have a bad ground
FWIW I think your battery is duff.
A battery has to supply AMPS at a specific Voltage
When you take a voltage reading, you get Volts at ZERRO Amps because the meter puts no load on the battery.
You said the Voltage at the solenoid was 12V with the key on ( almost no load ) then dropped to 0 when you turned the key to the start position ( high load )
That is the sign of a dead battery but before we advise you to open your wallet , I for one like to verify what you are spending your hard earned cash on will fix the problem.
I do not like waste.
Wasting water, wasting energy , wasting resources wasting time or wasting money,