How precise do I need to be when ordering a custom exterior door?
That's a valid concern.  1/8&quot; is a lot with this kind of thing. I'd want 1/16&quot; precision or better. I'd expect that if you give them locations to 1/32&quot; (not that difficult even with a standard tape measure) you'd be ok.
I would order a door without hinge mortises (or just buy a standard slab) and do that part myself. If you just buy a slab you can temporarily screw the hinges on, verify height and operation, mark the hinge locations and do the mortising work.
Depending on your hinge style, that may just involve shallow cross-cuts and a little chisel work, without any corner shaping. If you do want to fit round-leaved hinges, a Forstner bit would do a nice job there to get the radii cut.