That's like, the difference between one ship full of Pilgrims and every subsequent ship, or between one wagon train and every subsequent wagon train, or between the people on the Mayflower and their kids (who showed up in a society that already existed).

The people who draw the distinction generally don't like the pilgrims settlers because of what they did when they got here. In contrast to immigrants, who did not "settle" anyone or any land. You can see a lot of this in today's politics as well.

The land is still being settled today; eg. Trump allowing previously unsettled land to be destroyed for oil and gas extraction

You are correct in that this difference between settler and immigrant is a gradient, but outside of programming few things are simply yes/no

It's more than a gradient, it's like counting up ones to make 1,000,000. Which one is showing up to a small number and making it big, and which ones are showing up to a big number and hanging around?

Ah, yes, the “Daughters of the American Revolution” types.

somewhere that you claim is hunters/gatherers*

Just because the farming and ranching looks different, doesn't make it not farming or ranching