Developing and maintaining new gymnastics skills require that you train two types of movements: strict, strength-based movements and dynamic, swing-based movements.
The intention of your training session should determine your grip — whether that be the false grip or a neutral grip.
A false grip gives you both more control and leverage while on the rings because of the angle you create at the wrist (shorter lever). For this reason, you are stronger in this position.
And if the intention of your training session is to get stronger, you should perform strict gymnastic movements in a false grip position.
As evidence of this, olympic-level gymnasts are now performing their ring routines in a neutral grip position for the simple fact that it is harder. A major component of artistic gymnastics is to make the incredibly difficult look easy, which is why they’re starting to practice more and more routines with an open (neutral) hand.
In any movement or drill where you find yourself kipping (dynamic), use a neutral grip. The simple explanation is that gravity will make it harder on you if you have a false grip.
Dave Durante, 2004 USA Olympic Gymnast, wrote:
When you move through the bottom of a swing, the intention is that you’ll be at your absolute bottom position in order to create a fluid swing. Here, angles are the enemy — and gravity will eventually pull you to the bottom point, no matter how strong you are in your false grip. Using your false grip during dynamic movement will cause slack — leading to unnecessary stress, less efficient movement, and also potential injury in the long run.#TechniqueMatters Blog
Perform your strict, strength-based gymnastic drills in a false-grip position. When working on your kipping, or dynamic, positions in a gymnastics movement, use a neutral grip.
You can also practice and build up your false grip strength by practicing drills like this or this.