were separated from the rest of the U.S., and they ran as a independent country, so yes Lincoln Army did invade the South…
The fact is that there was never a
real nation anywhere on earth called the "Confederate States of America." No such nation was ever recognized by any actual nation, and the land that it pretended to be was always part of the United States. Lincoln deployed United States troops
within the United Staes in defense of the United States.
My Uncle Dave could declare that the island where he lives off the coast of Maine is the sovereign nation of Daveland, but he's not going to be recognized as such by anyone - except maybe his cousin Bert.
The legitimacy of any nation is established by the recognition of actual, sovereign nations. Pretending otherwise does not make it so.