Princes Street station wasn't quite on Princes Street, it was a little way down Lothian Road beyond the Caledonian Hotel as mentioned. Looking on aerial mapping you can see disused lines going west (now a road), south-west and north. The south-west one joins the line towards Carstairs just where it passes over the Suburban line north-east of Slateford station. The train to the south would have taken this route, then the current route via Carstairs, Carlisle, probably WCML to Stafford and Wolverhampton.
The connection now used from Slateford to Haymarket was (if I recall) originally a branch to serve the brewery, which was also connected to the ex-North British network near Haymarket and it was possible for trains to shunt through the brewery's sidings to get from ex-Caledonian to ex-NB tracks. When the decision was made to close Princes Street station, all the other routes out of it either linked to the NB network or were closed as well, and it was simply necessary to make a more useable connection to Haymarket so that trains to/from the south could use Waverley instead.