What screwdriver do I need for this ‘star’ shaped fitting

Looks like you need an 8-point socket. It is hard to tell the size from the picture but it looks small, you will probably need one in 1/4" drive. If you want a driver for it, you can get a 1/4" drive socket drive handle.

The loose end is for your index finger to press the screwdriver against the screw and thus provide the second fixed point for its axis while the thumb and the middle finger are turning the screwdriver. It allows the most precise operation of all manual screwdrivers.

I would also speculate that this might not be the right tool for the work. This kind of screwdrivers are for precision work where you need to be very exact in where you touch your device and exert a very limited force. You need a regular screwdriver, which would allow to exert higher torque and at the same time press stronger along the screwdriver/screw axis to prevent them from losing each other.

For the screw you want to turn, you need one of the bits in your second photo, in the holder with the word "Poholy" on it, number 5...7 from the left in the photo (4...6 if you turn it upside up). Take the one that fits the screw best. You will also need a handle, which is missing in your photo. It looks something like the blue one in this photo.

You can use the electrical screwdriver from Worx that you showed in another post, but choose the correct bit and as low torque as possible to start with. With the electrical screwdriver, it's very easy to damage the screw cap.