No expat means expat and immigrant means immigrant. Immigrant implies that the intention is to settle in the host country, whereas expat implies that the reason for staying is temporary and hence the intention is not to stay permanently.
For example, if I am doing a PhD in France, with no intention of staying in France afterwards, wouldn't it be rather inaccurate to call myself an immigrant to France?
You are not an expat in the country you are currently in. You are an expat in the country you left.
If you're in a foreign country, you're either an immigrant (as a generic term) or, if you want to get pedantic, you can use whatever more apt definition they have (visitor, migrant worker, non-immigrant, etc).
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.
This is the distinction between permanent resident and temporary immigrant.
An immigrant is someone who moves to a country other than their own with the intention of residing there for a significant period (usually at least one year), regardless of their ultimate intention to stay permanently or leave later.
You can just look up the definition of the word immigrant.