Push button switch terminology
I would call it "Alternate Action" if the button returns to the same position each time you remove your finger, but the switch position changes on each press.

With a latching button, I'd expect the button to stay partly depressed with the switch in one position, and fully out with the switch in the other position.

I don't think I've ever seen a switch with the delay function you describe.  That sort of thing would normally be done with the aid of a microcontroller or time delay circuit.