Both of the photos show grippers fitting outside the screw, but there is another way. My father used to have a flat blade screwdriver (I wish I had a photo, or better still the screwdriver) that gripped inside the slot.
The tip was divided into three parts. The two outer sections were integral with the tubular shaft, and formed a fork, similar to a security screwdriver, whilst the centre section had a straight shaft and parallel tip like an electrician's screwdriver, passing straight down the centre of the shaft. This was spring loaded to rotate about 60 degrees relative to the rest of the blade. To fit it to the screw, you simply turned a knob to align it with the rest of the blade, and when released it gripped inside the slot.
Brian