12.5 isn't enough to keep your battery charged. I still wonder why the battery died overnight - but you might have depleted it while riding and shut it off already too low to restart.
Sorry - I missed your earlier question about whether the rectifier is a part of the stator or separate. It is separate. It isn't uncommon for a voltage regulator to go out on any vehicle, though I have seen it more frequently in scooters than any other vehicle. Possibly because they are normally in an enclosed area and exposed to vibration... I don't think Kymco's regulators fail more or less than any other brands though.
Like MarioG said - the diode is the voltage regulator. Some people also call it a rectifier - I feel that what people call it depends on their area of expertise. On your Agility 125 it should be a small aluminum box with a black plastic 4 wire (red/green/yellow/white) connector on one end and a mounting bolt on the other. It should be under the front cowling - near to the horn. Any Kymco dealer should have them in stock - I would guess between $15 or 20 bucks. You can find an aftermarket replacement if you have a good automotive supply store - but I would try for an OEM first.
Its' nice that the Batteries Plus people will check your system for free - but unless they recharged the battery first the results are tainted. Sometimes people know how to run a test but they don't really understand it - or don't understand how to use it in a specific situation. Since your tester said there was something wrong with the stator and the diode - he was basically saying there is a problem with your charging system (the rotor and stator make up the alternator - and when you add in the regulator that is basically your whole charging system). When I work on computer motherboards I call it a diode, in 110/220 AC appliances I call it a rectifier, but on vehicles I always call it a regulator. They all allow electricity to pass in one direction easier than another - but when someone refers to them differently I tend to feel this is not their area of expertise.