The borders of Greyfriars cemetary were also much larger than where they are now, and again, a mass grave site.
Leith, of course, was also its own town, hence the pub 'The City Limit', w/one of its sides being in Leith and the other in Edinburgh.
Corstophine was a village, as was D-Mains, New Haven, etc.
New Haven was not considered part of Edinburgh when my MIL was a child.