These locations are along the line.
This terminus was in Musselburgh itself, on the east bank of the River Esk next to the 'Roman Bridge' (actually medieval). The site was fairly cramped being constrained by the river on the north side and works to the south. The station replaced Musselburgh [1st], about a mile to the south, which became Inveresk.
Drem to Edinburgh: Including Gullane, Haddington, Tranent, Musselburgh and Fisherrow Branches (Scottish Main Lines)