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 back swing and a long follow through. The advantage of this type of stroke is the shorter the back swing, the less chance of error.
The grip most certainly will influence the club head, 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 back swing.
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!
Good Grip (hands) leads to good wrist action
Wrist action accounts for 90 percent of bad shots. Why is this so common? The natural way to control the club is with the hands and any attempt to use the hands encourages breaking of the wrists. To explain wrist action: There are two directions the wrists can move in the golf swing. That is the forward/backward or up and down. I am going to refer to the forward/backward action as the breaking of the wrists. The up/down as the hinging of the wrists. Some teachers call the up/down movements cocking the wrists.
This is one reason that the grip is so important. A faulty grip encourages breaking. One does not have to learn this up/down (hinging) if the grip is correct, and the left arm is in the correct position the hinging will be automatic. . The legs moving forward while the club is still traveling backwards forces the wrists to hinge.
In order for this action to happen keep the left wrist flat in relation to the back of the left forearm and the back of the left hand. It's as simple as that. If the left arm is in the correct position the wrists will hinge, if not there will be a breaking motion. Don't try to cultivate an independent wrist motion. Let it be natural. Swinging the club back farther than shoulder turn forces the wrists into a cupping(breaking)position. Result: A throwing motion.
"Learning the Swing Center Feel for Better Shot making"
The simple way to learn the proper feel of your swing center is to place a golf ball on the ground and, with sun to your back, take your stance in a position whereby the shadow of your head covers the ball. As you swing, keep your eyes on the shadow. During the swing the shadow will move slightly, however if it moves off the ball you have lost the correct swing center. If the shadow remains on the ball you have kept the swing in the correct position. Repeat this excellent drill daily so as to reinforce good habits that will yield the results you seek.
You swing the club by feel, and you learn feel through good motion. Keeping your eye on your shadow will teach you the feel of your upper body staying in position - neither moving to the right or left- nor up or down.
To keep other body movements from moving the swing center keep your weight to the inside of your feet. Never allow this weight to transfer outside as this will pull you out of position. Allow the rotation of the shoulders and arms to carry the club to the top of the back swing. These are some key points that will give you feedback in understanding how to feel the proper swing center.
Learn The Swing on the Practice Range
To a beginning golfer hitting a golf ball seems easy until they try it. Golf can be a sometimes frustrating, and time-consuming game to learn. My recommendation is that players practice patience by not hurrying to the golf course to learn how to play. Do your learning on the practice range where your mind will not be on scoring, but on learning.
Some things to know and practice: First, and most important search and ask around about a respected golf instructor in your area, and get them to give you information on their program prior to starting an instructional series. You don't want someone giving you advice who is not qualified, and can impart poor information that can create bad habits.
Secondly, You learn to play golf by feel, not mechanics alone. Too much detail is confusing. You cannot think your way through a golf swing. You feel your way!
Finally, short practice sessions regularly are better than one long period. Practice, and good repetition, will teach your muscles to learn to feel - creating your own internal dialog from within that you, and only you, can describe to yourself.
Ultimately, start out with a plan, and take your time because as the saying goes "You must learn to crawl before you can walk."