I get that you can make the argument that they're merely making a semantic point. However, if that side of the debate actually agreed with us that these people shouldn't even be here at all, what difference does it make what we call them? If the side who wants them gone had their way, they'd be gone back home and they'd no longer be in any illegal status in any sense of the word.

It only matters what we call them, if you want to keep them here forever. I think the present-day recommended term is probably just "immigrant" right? So basically we should call them the same thing we call the people who waited years for their turn and proved that they had a positive contribution to make to our society.

The term for immigrants without papers is "undocumented immigrant". The largest group of undocumented immigrants are people who entered the country legally, and then overstayed their visas or otherwise violated their terms (usually by working on a tourist or student visa). This is a civil offense.