if it is like most briggs engines that i have seen the red wire is from the stator for lights and battery charging, and the black wire is the kill wire coming from the magneto
Actually, all of these are just starting points for analysis of how to fix the problem. Note that the 255470 model you cited (one of your manual links) has essentially the same circuit as I was describing. Same for your 255520.You can't assume that the schematics are the same because the model#'s are in the "ball park to each other. . .
You can look up the manuals for-
and the model#'s are "closer" than 255560. You would also see that there are at least a couple different schematics.
One extreme case in point is- 917.
IF both engines were Briggs opposed twins, the most sanitary method would be to simply swap the alternator & flywheel from the old engine to the new one.