There are rich arab or chinese expats in the west, are they called immigrants? I think expats refers to the temporary/revocable/definitely-not-forced status of the immigrant. In my greek native, expat and immigrant are used interchangably (ομογενής /μετανάστης)
That said, "expat" is the wrong term for someone who renounced his original citizenship. I guess they should be called "pats" now.
In the US context, Mexicans who come to the US for work, with plans to return to Mexico after 5-10 years, are called immigrants. Thus, "immigrant workers" not "expat worker" and "undocumented immigrants" not "undocumented expats".
The same is true in Europe. The places in town where North Africans, or Iraq, or Ethiopia live are often called "immigrant communities", not "expat communities." Even when the population has no plans to return, or are even native born.
While on the other hand there are many British people who retire and live the expat life overseas where it's warm and cheap, and don't plan to return to the UK. Or Arthur C. Clark, a British expat in Sri Lanka who had no plans to leave. The Sri Lankan government created the category of resident guest status in 1975 so he could stay.
This strongly suggests that 1) your understanding isn't correct, and 2) there is a social class distinction between "immigrant" and "expat".
That said, "expat" is the wrong term for someone who renounced his original citizenship. I guess they should be called "pats" now.