You're quite right. What we are (or were) is a nation of settlers, that has successfully absorbed several waves of immigration.
The current massive population transfers from the 3rd world are not an immigration wave because, under the 1965 Immigration act, there will never be an end to the influx - and waves have ends. With no end to the influx, there will be little assimilation or transfer of loyalties from old country to new. So what today's immigration practices are really is nothing less than the phased abolition of the United States.