This is silly. I migrated to the US from a wealthy country and I am an immigrant. I've also lived temporarily in other countries, and therefore was an expat.
Well yes, going by the dictionary definitions that's true, and I commend you for adopting the immigrant term. I'm an immigrant too, and proud of it.
At the same time, it's also clear that the status of the country determines whether people call themselves immigrants or expats. In the US people are immigrants, in Japan they're expats, for example. US citizens in particular very rarely refer to themselves as immigrants in my experience.
It's not hard to understand.