the reason that there are two wires on one and only one wire on the other is that they have different Charging systems.
If the engines are similar engines, the easiest thing to do is to take the charging coils (stator) off of the old engine and out it on the new engine. Then you won't have to change any of the wiring.
If the stator on your old engine is bad, then you should get a new stator that has the same single wire system that you had on the old engine and install it on the new engine.
there is also the possibility that the aold wngine has a diode built into the wire comming from the stator and the two wire may not. This could cause major trouble.
If I was working on it in my shop I might be able to find some way to convert it, but there isn't any published way to do this that I know of.
The only way the engine would have an electronic ignition on the side is if someone had replaced the points and condenser with an aftermarket electronic ignition module.
In that case, the new engine will have only one wire coming from the engine Ignition and it is the shut-off wire that goes to the ignition switch to chut the engine off.
I am a little confused, because there is also either a set of wires or one wire that is the charging system wire that charges the battery.
WE have to make sure that we are talking about the same wires.
You have to make sure that the wire you are talking about actually goes to the ignition that is under the engine cover. That wire is the the shut-off wire.
Be sure that the wire you connect to the ignition switch shuf off is NOT the charging system wire. It will short things out.
If you could take a picture and post it here, I could easily tell you what is going on. More than one picture would be great.