If I understand the diagrams, the yellow wire from the supplied Alt plug ties into the pos from the battery after the fuse.
The Yellow (or is it Yellow w/white?) wire needs to be connected to full battery power. Most diagrams show, and most people do, the simplest way of achieving this which is to connect it right there to the main charge post on the alternator just a few inches away.
The best practice, and what Ford does, would be to get the wire as close to the battery as possible.
Sounds like your existing Painless Yellow wire is routed back into the harness and would still be a good source of power for this wire. Make sure it's a good tight connection so it gets an accurate reading of the battery/system voltage.
Which leaves the green and red wire from the supplied plug. Would that just tie into the green and red from the harness and I’m good?
Yep, simple as that.
It only needs to see 12v power when the key is ON and turn off with the key.
If your alternator charges as expected even without the White w/black wire, you're good to go. If it does not, you'll need to get a new plug where all three wires are still intact.
The other end of the White wire connects to a small blade connector on the side of the alternator.
Maybe the Explorer is the one that does not need it? I should know since I've wired several of them. Just don't remember at the moment.
(edit: I see that's one of Ryan's old diagrams specifically for the 4G and does not show a stator wire as needed. So you're likely good to go with just the Green and Yellow wires)
Paul