Modern Bud Jones type inlays are produced on a self adhesive material.
When they fall off or are removed, invariable the adhesive remains on the back of the inlay with zero residue on the chip.
Consequently, if re-applied, there will be nothing that creates any imperfections on the surface regardless of orientation.
After all, why should it be noticeable, the inlays were applied by hand when the chips were manufactured.

I would say Ive encountered maybe 50 to 100 BJ chips where the inlay has come off or is at least raised around the edge to a degree one could slide it off (they don't really need peeling).
Seems to me that extremes in temperature is the major reason they come apart. When I have boxes shipped to me, they can experience extreme low temperatures in the aircraft hold which seems to freeze the adhesive (rather like putting something with chewing gum stuck on it in the freezer), and I have encountered chips in packages received where the inlays have slipped or come off, and the adhesive is back to a very sticky state as the chips start to get to room temperature again. Re-applied at that time, they will once again stick perfectly.
I believe you will find that warming the chips, possibly just by holding in ones hand and rubbing your thumb on the inlay for long enough will cause the same thing to happen. No exacto knives or other tools required.