No, the difference is whether you are just residing outside your home country or actually immigrating to the new country. It is the difference between a vacation and moving somewhere. It is more along the lines of external patriot than former patriot.

Someone who still sees themselves as a citizen of their home country and just happens to live elsewhere is an expat. So an American living in Mexico is an expat, no matter what their length of stay is. If they immigrate, they are moving permanently and they see themselves as a part of the new country, either by seeking citizenship or claiming that as their ‘home’ as part of their identity.