Originally what we now call ‘Back Bay’ was called Front Bay. When the rail road was put in in 1871, it cut the bay in half. The portion of the bay that still had lake access remained Front Bay and the half other half was called Back Bay. I don’t know exactly when- but Back Bay was filled and became Foss Field. I assume it’s around that time that Front Bay started to be called Back Bay.
Now, google maps labels the area as Back Bay. I wonder if this name sticks better because first of all- it’s logical, the bay is on the backside of the center of town, and certainly comes second to the awesome Lake Winnepesaukee, and it is also catchy to the many people who are familiar with the Back Bay of Boston.
PS mad appreciation for Brewster Academy being labeled in huge letters on the oldest map :p
EDIT: I’m not sure if my assumption is right. I will update again when I know, and I will find out…