Both those places don't seem to have ever been on my radar, so I can't help with names.
Tom Kitchin's place in Leith is well worth a visit. Ditto malcolm Duck's. The witchery in the Lawnmarket has great surroundings, but the food can be variable and is very expensive, their other place at Prestonfield House Hotel is good. The restaurant in the Museum of Scotland is absolutely amazing.
ETA. How could I forget to put Martin Wishart! He and TK are both Michelin-starred restaurants. Crivvens, see how posh The Shore, Leith is nooadays?!