You are an emigrant in the country you left.
Immigrants and expats are overlapping terms, but there are some differences in common usage. An immigrant may intend to stay in the host country permanently and make it their new home, while an expat just happens to live there for the moment, often for a specific purpose. Expats are often more privileged than immigrants. An immigrant may seek to integrate into the society and the culture, while expats are more likely to have a purely transactional relationship with the host country.
I agree with your characterization of expats as typically being more privileged, and the context you give.
That said, it is expat and emigrant that overlap, not immigrant and expat.
An expat is someone who has left his country, with varying degrees of permanence and volition. Eg: to be exiled is to be expatriated by force.
Absurd example: you could technically be an expat without immigrating (or visiting) any other country if you kept to international waters (or unclaimed space?).
The distinction between "expat" and "immigrant" is made by governments as well. Example: as someone who over time held multiple non-immigrant and dual-purpose visas, became an immigrant (permanent resident), then naturalized, I can tell you "expat" was nowhere to be found on the dozens of INS and USCIS forms I've had to fill out. It was never a status I could apply for or one I was given. On the other hand: the US embassy, and my home country's embassy, do offer me "expat services" where I currently live.