Yeah I agree with all of that, so they don’t actually assign different polarities but they talk about them they opposite way, like saying the negative terminalI don't know of anyone that knowingly redefine voltage, but the way they use things it is often the case that they would need to in order to be internally consistent (i.e., be able to dispense with the magical mystery minus signs).
Saying that the negative terminal of the battery is at a higher potential than the positive terminal is simply wrong, unless you redefine the laws of physics to result in the polarities of all of our batteries (and every other voltage) to swap. This is perfectly valid, by the way, since the choice was arbitrary to begin with. But they won't even attempt to go there.
Saying that the negative terminal of the battery is at a higher potential than the positive terminal for electrons is a move in the right direction, but it still ignores that electric potential is a property of space (assuming a conservative electric field, which is reasonable for this discussion) and that it is independent of whether the charges in that space are positive or negative (assuming those charges are having negligible effect on the electric fields in that space, which is usually the case for practical circuits).
What they should say (and some do) is that, owing to their negative charge, electrons have a higher electrical potential energy at the negative terminal of the battery than they do at the positive terminal.
Is higher potential. I always heard that the word electric potential was defined as J/C so a higher positive electric potential meant higher energy for a positive coulomb, therefore implying opposite for negative coulombs(electrons).