Of course our experience of our praying spaces is always “something else”!
Used to be our assembly houses were known as exactly that - assembly houses, batei kenesset. They were also referred to, in more reverence, as kehillot kodesh, holy communities. The Greek word “synagogue” (συναγωγή) means “assembly,” and we have adopted that word as well. Of course, “congregation” implies the same coming together that “assembly” would indicate, and indeed all of the bodies mentioned in our title have “congregations.” Back in the day - say, 2,000 years ago - the president of the synagogue congregation was the Latinized “archesynagogus.” (Our current president might prefer “archesynagoga.”) Also back then, the ancient Temple in Jerusalem (the Beit Hamikdash) was still operating, and synagogues served as remote meeting and praying places.
After the destruction of the Temple, synagogues were sometimes called batei tefillah (houses of prayer) or batei midrash (houses of study). That latter term may have been the impetus behind calling the place a shul (שול), the Yiddish word for school. This term is still more common in Orthodox congregations, though it spills over into other denominations as well.
As the Reform movement took hold (after the principles were agreed upon at the 1885 Pittsburgh Conference which was actually held in Allegheny City, not Pittsburgh, but I digress), the notion of using the term “temple” as an intentional statement that the Temple itself is being restored, which had begun in Hamburg, Germany, in the early 1800s, became solidified among Reform congregations. Since then, some Conservative congregations also adopted “temple,” more with the notion of being small Temples (in Hebrew mikdash me’at (מקדש מעט), an actual rabbinic term), replacing the goings-on at the Temple with prayer and teaching, rather than being a part of the whole reconstruction of the Temple. That somewhat tentative usage may be walking the fence between the requirement that the Temple be rebuilt at the time of - or some say in order for - the Messianic coming, and the acceptance that we are actually replacing the Temple with something else.
So the word “synagogue” serves us well, as it reflects our multi-purpose use of our buildings nowadays, for praying, studying, meeting, celebrating, observing, taking action, and many other purposes. And it doesn’t really refer to any one denomination of Judaism. Note, though, that generally we use it to refer to an actual building, thus leaving those congregations which meet without four walls and a roof to use the word “kehillah,” community, which we also use from time to time to refer to the assembly of congregants inside our synagogue.
Sources: JewishBoston.com, Judaism 101, Wikipedia, Encyclopedia Judaica, general knowledge.