Ok. From what I gather from the reply so far, it is not a short. It
seems the charging system is broke somewhere. I replaced the
regulator with an automotive one, as I could not find a lawn mower
one handy. I will start there again to check.

I am looking for information on the charging system of a Briggs and
Stratton (I do not know if it will be any different when installed on
a Toro). I do not know if getting one of the Briggs repair manual will
help me on that. Can you guys point me to the right direction?