Yep, I do have air. The PO said it didn't work. So this might red wire be
in relation to that.

But I'm still mystified as to where the sensor wire would then hook into.

I would think it would be a female spade recepticle, however I couldn't
find one.

Also, the AC connectors I'm pretty sure are correct.

Does your brown sensor wire connect to a Blue / Black wire then ?