At my U.S. Tower/approach control, we used half strips for IFR flights, no strip holders. For VFR pop-ups, we used to use paper strips, then just scratch paper, neither very satisfactory to me. Eventually, we had pads made with outline of five strips printed on them. Small, neat, orderly; worked well.
And I believe the time stamp at the NY airports was an electronic system to forward departure times to the Tracon some miles away.