Normal welding needs heat to melt the metals. Cold welding happens without heat. Two metal parts will cold-weld on any smooth, touching faces if the air molecules that keep the two separated disappear.
Your description, and wiki's, both sound like it is welding.
And after spending an embarrassing amount of time last night going down a welding rabbit hole, I have not seen anyone else claim that "cold welding is not welding". I'm pretty sure it is.