I've never seen a case where an card wouldn't work in another bank's automated ATM machine. I know this used to be the case, but there are a bunch of logos on the back of my card, and a bunch of logos on various ATMs, and as long as there is an intersection of at least on of those logos it should work.
And there is a big difference between giving basic proof of ID information to a regulated financial institution, vs. handing it over to an unknown entity.