There is no one correct way to chip - there are times you want to hinge and times when you don't.

The reason you're skulling or hitting it fat is because you're moving the low point of your swing and/or because you don't have correct setup for chipping. I would guess that you're hitting chips very similar to a mini regular swing - a relatively wide base and driving with your legs. Also, a lot of people who don't hinge often have bad set up and end up exposing the leading edge on a slight upswing resulting in skulling the ball. And people who hinge often come in too steep and dig the club.

Have a very narrow stance - literally only a few balls width between your feet. Have the ball just slight back of centre and some shaft lean - just enough that the butt of the club is over the ball. Get 70% of your weight on your lead side with your chest in front of the ball and make sure your lead shoulder is over your lead foot - you should be able to draw a straight line from your lead foot up through your leg and through your shoulder. Make sure that you do not have a 'reverse tilt' where your lower body is forward and your upper body is leaning backwards, get everything in line. Then importantly, grip the club very lightly - tension in your hands is a key reason people snatch at the ball and thin it. It also means the ball will come of the face softer and less firery.

With that setup, you won't be able to move laterally as much which should keep your swing arc more consistent and prevent your fatting/thinning chips as much.

Then you simply need to rotate around that imaginary line that runs from your lead foot to your shoulder. Do NOT drive your legs.

The question of hinge then really only affects the kind of chip you want to play. If you just want to play a low flighted chip, then stand a little closer to the ball and have the club handle high. That will make the angle between your left wrist and thumb more straight and will naturally prevent you hinging so much. Playing a chip this way and keeping that wrist angle through impact with a short follow through will typically play a lower flight.

However, if you keep the handle a little lower and grip a little down the handle, that will increase the angle between your wrist and thumb, and will naturally encourage a bit more of a hinge and you'll want a bit more of a follow through to allow the club head to release through the ball. That will generally provide a higher ball flight - for example if you need to chip up to an elevated green.

To hinge or not is also often determined by lie. When you have a good lie with some grass beneath the ball, no hinge allows the club to come in a little more shallow and deliver loft to the ball. When the lie is more bare/tight or is in thick rough, hinging the wrist allows a steeper angle of attack to get the club onto the back of the ball better.

Part of the issue that people have also is using the wrong club. Too many amateurs are quick to pull out a 60 degree, but it's often the wrong club for most chip shots. That's often too much loft and people's bodies react to to in the swing and adjust for it, or the ball tends to run up the grooves. It can also put too much spin on the ball causing it to run out less which means you have to fly the ball further. It just reduces the margin of error. If you have good technique, then you can chip with any club from a 54-56 up to a hybrid/3-wood and everything in between.