Go ahead and get Tacky on the Greens
The shove type: The word shove is misleading. An actual shove is illegal. It's only a feel. What gives this shove feel is the stroke is made with a very short backswing and a long followthrough. The advantage of this type of stroke is the shorter the backswing, the less chance of error.
The grip most certainly will influence the clubhead, but any type of grip that will keep the wrists from opening or closing the blade is a good grip regardless of how the hands grip the club. Whatever grip is chosen the thumbs should be on top. The thumbs are the feelers and being on top one can feel if the blade is square. Any stroke that holes the ball consistently is a good putting stroke. Regardless of the style of the stroke the putter head should never be taken to the outside of the target line on the backswing.
Finally, I have a drill that may work for you when practicing your putting stroke called the Tap stroke. To learn this stroke pretend there is a tack stuck in the back of the ball and when you make your stroke have the putter head drive the tack further into the ball. This will encourage you to hit the ball with more authority which with this shorter stroke there will be less chance of the putter blade getting out of what is called square position. So start getting "Tacky" on the greens!
Please contact me at firstname.lastname@example.org if you have any questions about this wonderful game.