In this video I talk about the 2 different ways you can catch a disc. The key is to practice catching and figure out what way works best for you.
I like a variation of the pancake catch more, where you keep the thumb of your lower hand pointing straight up and you use the upper hand to guid the disc into your lower hand (fingers and thumb). You do this by holding your upper hand at about a 45 degree angle (fingers pointing upward, 45 degrees) and slightly in front of the thumb of your lower hand.
The advantage is that the disc cannot go through your hands and that you are catching it further away from your body. Also, as you explain, you have less problem with 'bouncy hands' while running as you can keep your hands closer together than with the pancake catch.