Did I make a mistake getting a 2nd cat?

It sounds to me like your cats mostly get along, and some of this is your worrying the situation is far worse than it actually is. Here's my read on some of their interactions:

• The cats cuddle and groom each other, though you only see them do it "a little." This is a very good sign that your cats actually do get along. For many if not most cats, merely being peacefully near each other is the most socializing they will ever do when they aren't playing. Grooming and cuddling is a level of socializing above and beyond what you can reasonably hope for when introducing cats that have never met before.
• The one cat lies next to the other, and the other cat gets up and leaves. This is also a perfectly fine interaction. The one cat leaving is sort of a polite "no thank you," in terms of cat interactions. Perhaps the one cat is more cuddly than the other, but that is okay, as the other cat is politely turning it down.
• After breaking up what you perceive as a bad play session, the "victim" simply walks away. If the bad play session were truly serious, the "victim" wouldn't merely walk away. I would expect it to flee and hide, or show some other signs of fear or aggression.

However, that said, I do think you may have a little bit of a problem with the play sessions. It's more typical for play sessions to be silent, though some cats might have the habit of vocalizing some during play without it particularly meaning something bad. Definitely it's a bad sign if they hiss, scream, or do quite a lot of vocalizing. The other bit is that occasionally the one cat runs and hides from the other. It's hard to judge when I can't see exactly what is happening, but my guess is that your one cat is sometimes getting a bit too rough for the other.

In these cases, it is a good idea to separate them. But instead of clapping, it might be enough to distract them using a toy, though if toys don't work, the clapping seems good enough.

You also mention that this happens in the morning, which is also common. Many cat owners feed their cats first thing in the morning, and cats tend to get hyper around mealtimes, plus they've just spent all night not being entertained. Therefore it might also help if you could change something about your schedule so that all this doesn't coincide into making your cats maximally hyper. Here's a few things you could try to see if any of it works:

• Change their feeding schedule so that they get their meals later, and not immediately after you wake up, and play with them beforehand.
• If you feed them dry food, you could also try using an automatic feeder to feed them while you're asleep.
• Keep them more entertained while you sleep. An easy way to do this is by giving part of their meals in food puzzle toys left out overnight, for instance, as long as you aren't concerned about one of the cats eating too much while you can't monitor them.

But since overall their relationship seems good, I don't think this is something to be overly concerned about.