There is an edge piece for this but we bought a metal finishing cap that matches. I'm still not sure if I should end them where the upper cabinets end or where the wall ends.

just try to do the neatest finish you can. is it a Schlutter® edge?

make certain it is level so it doesn't add another line to draw the eye.

I hope you are running the tile vertically and not horizontally as shown.

Why do you think it needs to go vertically?

End the tile run to be in line with the upper cabinet. Finish it off with a trim piece.

An overall pic of the entire kitchen will help with this decision. Context is key.

Hey there - Your instinct is right! A strip of bare drywall next to tile usually looks like a mistake but in this case, the space looks so narrow between your uppers and the end of the wall (door side) and on the island, that I'd recommend extending the tile to the end of the wall; it looks to me like it is less than 1 inch of drywall to your door.

id go with tile backsplash to end of upper cabinetry. the amount of wall between upper cabinets and door is negligible

Use Schluter edge

End it where the uppers end, that's what my very experienced tilesetter recommends.

And I wouldn't install the tile vertically. They look to be 2x8 tiles, which is going to be awkward if the space between the upper cabinets and the counter is the standard 18". You'll have little cuts left over at the top, which are going to look bad.

Agree with the advice here about ending the tile run in line with the upper cabinets — that's the cleanest line for the eye, and a metal edge cap like you're using will give it a really finished look. If you haven't already, dry-laying the tiles before you set them is worth the extra time so you don't end up with any awkward cuts at the ends. It's going to look great once it's up!