Regimented Catching Practice
Just like throwing, catching comes from practice.
How often do you see someone throwing before a game focus really hard on the throw and not on the catch? You'll hear this player swear and scream when they miss a throw, and then catch the throw that came back with one hand while joking with the person next to them.
On Z, we used to do the a throwing medley at the start of every practice and every game. This throwing drill consisted of 50 throws and 50 catches for each person. It involved circling so that you saw every wind. That was more throws and catches than most players had during the rest of the day. Everyone on the team knew this because the captains reminded us. So, everyone did it and focused on it. During that medley I made sure that I leaned in to catches, leaned and stretched in, used two hands when it was right. If you were lazy, someone yelled at you. So, you caught the disc. This works for some and not others, but it worked with that team for years.
As someone who primarily played near the disc as a handler, I definitely felt that I was a little lucky in the area of catching. The difficulty of catching dumps and swings is definitely lower than that of catching come-backs. The two most critical components of the catch were focus and positioning rather than skills with the hands. Body positioning, for me, primarily consisted of making sure that I got my body between the disc and my defender. This enabled me to make the catch that I wanted to make. I'd work on catching while leaning in, either with the claw catch or the pancake. I practice that in the medley and I used it in the games. Dump and swing throws were usually targeted for around the shoulder height, because they are tougher to D. Again, something to focus on.
For me, I thought of catching as fairly regimented. If the catch was between the knees and eyes, then it was a pancake. If it was over the eyes, it was two hands (claw), usually with one hand stopping it while the other grabbed it. If it was lower or a layout, than I worked to get 4 fingers instead of a thumb under the rim. That's how it was going to be caught, so that's how I practiced it.
The other thing that I thought was important, and especially important for a handler was the ability to turn a catch into a throw extremely quickly. Again, practice it.