I see references on the internet to the tunnel having the name "Easter Hailes Gate Tunnel" - and on OSM the path in the tunnel has this name. If you're looking at a direction sign I'd caution that those making up these signs don't generally use any different information to what we'd use... so they shouldn't be seen as adding much authority. It may even be that names on a new sign are put on it because of what's already recorded on OSM... a circular process. Water of Leith Walkway is indeed a relation: https://www.openstreetmap.org/relation/31454