Back to the electrical system for a moment...
I must respectfully disagree with Rivets regarding his comments about the regulator/rectifier block and the diode with heat shrink over it. Regarding the presence of the diode, I'm pretty sure that's the line which goes to the magneto. I know some charging systems utilize a series diode for simple rectification but I don't believe that's the case with your engine. The diodes are used in series with the magneto primaries. They prevent flyback from the magneto primaries from damaging the reg/rec block when you shut the engine off. L120s do use a reg/rec block as you have on your unit. The stator should have two black wires going to a connector that plugs into the connector with the two yellow wires of the reg/rec block. The stator plugs directly into the reg/rec connector through no intervening connector(s). The mating connector to the single red wire coming out of the reg/rec block likely has two wires in it. One goes to the Ign Switch and then to the lights and the other should go to a fuse and then to the battery.
I'm a bit confused what current you were trying to measure, given your reg/rec block is not connected. Normally to test the charging system, you run the engine at high throttle and measure the stator voltage. If that's OK you then measure the current coming out of the reg/rec block to the battery. If that's OK, you know the reg/rec block is good. Not sure what you tried to measure given that they are disconnected.
Regarding your measuring charging current as the Briggs manual spells out, under "normal" conditions, you would unplug the connector of the red wire coming out of the reg/rec block. You would touch your red meter led to the red wire coming out of the reg/rec block and touch the black lead of your meter to the POSITIVE(yes, positive) battery terminal and that will tell you the charging current. Your meter must be set up to read DC current on the 10Amp scale and THE RED LEAD OF THE METER MUST BE IN THE 10A JACK OF THE METER. NOTE: In your pic, the meter is set to read 10A DC as it should, BUT you have the red lead in the wrong jack. It must be in the leftmost jack to measure DC current on the 10 amp scale. DO NOT TOUCH THE BLACK LEAD OF THE METER TO GROUND OR THE NEGATIVE TERMINAL OF THE BATTERY WHEN PERFORMING THIS CHARGING CURRENT TEST. If you do, your meter will act as a dead short and the fuse inside the meter will blow. Remember to move the red lead back to the right jack to measure voltage!!!
... I hope this helps a bit