I've only been here about 4.5 months, but I already feel the same way. This is my home. I've never felt any deep connection to any city I lived in in the US, but I do here.
I still, occasionally, use the term "back home" to refer to that place from whence I came, but my husband dreams of someday going back to the States, so he readily calls it home (even though he only ever visited, and has never lived there yet). I don't seem to be able to make him understand that I'm not opposed to going back after he finishes uni in a few years, but I'm only just settling in, and I don't like thinking of Arizona as home again already. It makes it feel like I'm only visiting here, like I don't quite belong in England, and that no matter how many years I'm here, I'll always just be a tourist of sorts.
He doesn't do it as often, which I appreciate, but it still really bugs me every time it happens.
Anyway, I feel your pain. I hope after you express to them in the terms you put here, they have a better understanding and make a better effort to change the terminology.
