Exactly the same way.
Leave the clip connected to a good ground.
Anything you touch with the probe that is live will make it glow & beep ( you have a good one ).
So you start the probe at the solenoid terminal then go up the red wire and push the probe into the back of each side of the fuse holder.
Again it should beep.
Follow that wire up to the ignition switch and poke the probe into the back of every wire in the terminal block.
With the ignition off only one wire should beep ( the one from the fuse ).
Get a strong magnifier and look at the back of the old switch.You should see tiny letters
G = ground
M = Magneto
B = Battery ( should be connected to the fuse )
S = Start
A = Alternator
L = Lights ( fuel solenoid is generally connected to here )
Some will have 2 L and or 2 A terminals
Now back to the mower.
All you are interested in right now is the cranking circuit so push your probe into the back of the S terminal while holding the key in the start position.
You can do it to all the terminals in the off, on and start for the heck of it if that will make the actions of the switch clearer.
Follow the wire from the S terminal to its next switch, should be the PTO ( Switch if electric PTO , safety switch if a lever PTO ).
Here you will need 3 hands because you need to test the wires with the ignition in the start position and the PTO both on and off.
You will find it easier if you pull the plug on the ignition switch off and poke a short jumper between the B & S wires on the plug.
IF the pto switch passes power through it in the off position, follow the wire , It should now go to the parking brake.
Same proceedure.
Mechanical gear box mowers will also have a reverse safety switch in the cranking circuit, hydros don't.
Eventually the wire will come back to the solenoid.
Some of the safety switches will have 4 or even 6 wires , 2 will be coloured and the rest will be black, brown or white with or without a stripe.
All the similar wires will be part of the magneto cut out and are not part of this test ( they will either be ground or open )
On some mowers the wire changes colour every time it goes through a switch while others it will be the same colour from the S terminal on the ignition switch all the way through ( usually yellow or orange )
Keep a note pad with you and write down the colours as well as weather it was live ( beep & lights ) when the switch was on or off.
If a wire is broken where the terminal crimps on and you poke your probe into the back of the plug, it might cause the broken wire to make contact which will crank the engine if you have jumped the plug on the back of the ignition switch,
Be ready for it as it can suprise you and do not put anything on top of the engine nor lean on it.
Somewhere from the red wire on top of the starter solenoid to the orange wire which triggers the starter solenoid there is a broken connection.
Double check that the orange wire does not have 12V on it when you turn the key to the start position.
If the orange wire goes live when you turn the key the solenoid is the problem.