Emms, I think you are right. Looking for something completely unrelated in historicaldirectories.org, I checked the earliest (History, Directory and Gazateer of Cumb & Westmorland, 1829) and on p154 there is a list of streets and squares of Carlisle with the following;
Water Lane, Court Square
Watergate, Water Lane.
In another bit, it says the railway to Newcastle is under construction but I seem to remember that Carlisle had 3 different stations (N/S, E &W) before they built the present one.
bob