OK, thanks for clarifying. I guess the last line I wrote is indeed the part that is non-obvious. I suppose it doesn't hurt to leave my answer up here, in case it clarifies things for others.

It looks like it. I mean, it is the natural thing to do, in my opinion.

I'm assuming that $\Delta$-generated means that a space is a colimit of the canonical diagram of spaces in $I$ over $X,$ in analogue with what compactly generated means.

@Qfwfq: It's equivalent to yours for Deligne-Mumford stacks. It's a standard trick that you don't need to check for all etale maps from a scheme, but only on atlases in this case, and then it boils down to exactly what you wrote.

Good point. I did say at first glance. At second glance, good point ;-). What would be reasonable for the case of Artin stacks?