Should I try to cut out so both the existing and the new drywall will
touch half of the metal stud on each side? And put in screws on both
Yes that is an effective way to do it.
Another way is to add a board, 1"x4" boards work great, as structure to screw to.
Clean up your hole so the cuts are straight and at 90 degrees to each other so you can use a nice square or rectangle piece as a patch. Place the boards so that half of it is behind the old sheet rock, length wise along the cut, and half is sticking out so you have something to screw the new piece to. Screw the board in place through the old existing sheet rock, make sure your screw heads are counter sunk.
Now you can cut a piece to fit the hole and screw it in place. Tape and mud the seams. Sand and mud, sand and mud and sand, prime and paint.
Floating a but joint with mud is an acquired skill.
Watch some YouTube video's and practice on a mock up of a but joint.