In my experience people will use "immigrant" to talk about were they're from by referring their nationality (i.e. "I'm a Portuguese immigrant") or explicitly adding a "from" and then using the country name (i.e. "I'm an immigrant from Portugal").

If talking about where they're an immigrant in, they will explicitly use "in" (i.e. "I'm an immigrant in The Netherlands").

Even though "emmigrant" is about where you were born and aren't living in anymore and "immigrant" is about were you went to, in my experience emmigrant is only ever used when physically in one's country of original and talking about living elsewhere (i.e. when in Portugal I would say "I'm an emigrant" whilst when in The Netherlands I would say "I'm an immigrant").

It's funny since as I'm writting this I remembered that when I first left my country of birth to go live abroad it actually took me a while to figure out the proper usage of the whole immigrant/emmigrant thing.

As I said, I was an immigrant in The Netherlands and worked often with other immigrants from all over there (mainly because until I learned Dutch I could only work in English-speaking environments and in my area - software engineering - those attracted immigrants), and most people would use "immigrant" when talking about were they came from (i.e. "I'm a French immigrant") and I only ever heard expat used instead of immigrant by people from Anglo-Saxon nations, overwhelmingly Brits and Americans.

That said, "expat" was used as a single word combining both "immigrant" and "emigrant" - in other words, unlike with the immigrant/emmigrant pair, the single word expat is valid both when one is physically on one's country of origin and when one is physically in one's host country: when I lived in Britain I did hear Britons saying that they were "expats" and meaning it as "living elsewhere than Britain".

And yeah, 2nd generation don't call themselves expats, but they also don't call themselves immigrants. It's only people from outside talking in general about people who are the direct descendants of immigrants in a country who will use "2nd generation immigrants" for the groups as a whole. Calling somebody who is a national of that country and has immigrant parents "an immigrant" in that country is only ever used as an insult by Far-Right extremists.