The question "which town is this?" has a technical answer (Boothbay) but also a "common" answer. For example, Springvale Maine isn't even a town, it's part of Sanford. But if you go there and ask anyone what town they're in, they say "Springvale". It's also what they will give as their address. The mailman doesn't care, they only care about the zipcode.

I think East Boothbay is the same situation. One of the ways I've checked this remotely is by looking at business websites. Your edit includes "Washburn and Doughty Associates" which has a website: https://www.washburndoughty.com/. The website gives the address "7 Enterprise Street East Boothbay, Maine 04544" (nice that it confirms the zipcode) and I think this demonstrates the "common" answer to "which town is this?". Maybe check a few more businesses before calling an anecdote a conclusion.

It's fine to have the addr:city tag be either the "technically" or the "common" answer, I've been trying to use the common answer in my work thinking it will make the map more relevant to locals and visitors.