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Are these (see the arrows on your pic that doctored) what you are referring to as "boots"?
View attachment 16297
So the bullet connectors are inside of those (sleeves, boots, whatever you want to call them).
Carefully pull the wires to separate - you don't need to cut those off. They are just there to give you a good way to (store) the wires on the original switchbox so they don't get pinched, chaffed etc.
If your new switchbox doesn't have the molding in the (case) to snap them in, then just leave them off.
That "three way" wire is the stop circuit - has a single feed from the switchbox and the other two wires go to the on/off switch on the lower cowl and the second goes on through to the stop button at the end of the tiller handle. When you switch "off" or press the button in the end of the tiller, those wires will send the switchbox power "to ground" to stop the motor.
That three way (plug) is downstream of the bullet connector that attaches it to the switchbox, so should not be an issue when hooking the new one up. You need that (circuit) or you won't be able to turn the motor off....
Graham is dah man!
Jeff