It looks like most of what I suggested will still work and there's some other options on there as well. I can't think of anything at Peters Yard that would be strictly vegan so you can probably skip that. Any nice cafe will have soya milk available though it's really uncommon to see anything else (my mom avoids soya so sometimes brings her own almond milk out).
Watch out for pre-made sandwiches (though I know usually bread is out, just in case) or ordering baked potatoes. They tend to automatically put butter on everything here so it's just something to keep an eye on.
Just returned and wanted to update - we ate at the following places which were phenomenal
David Bann (vegetarian - and really really great, even the omnivores in my group loved this) Bia Bistrot (not vegetarian, but lots of options). Everyone said this was one of the best meals there The Indian was really good.
In terms of running routes, the best run I had was starting early in the morning at Edinburgh Castle, down the royal mile (very scenic), around Holyrood park (following a 10K route I found online), and back. Very very scenic, lots of historic urban scenery and green space/krag scenery too. There were lots of opportunities for stair interval diversions too!