@neophyte365 The only time I had an issue with a door pull-up bar was when I was traveling, put it up in the hotel doorway that went to the bathroom. Did a pull-up and hit my head on the ceiling, didn't realize how low the ceiling was from the frame!
The ones that sit on the lip rely on torque to stay in place. Most things with calisthenics are all about balance, even pullups and levers. You shouldn't have to worry about pullups or even going inverted if you're slow and controlled, because your body will want to be balances directly under the handles. It's only if you're going crazy, forcing yourself down to do pullups faster, could you accidentally lift the handles and cause the bar to come undone from the frame.
@neophyte365 I used the same basic Walmart pull up bar for 15 years. The only reason i stopped was because my new place had doors with trim that was too tall to allow it to fit properly. I moved on to one of those standing towers that also has dip bars and push up handles built in to it. I definitely prefer what i have now. But it's the difference between $20-25 and $150-200ish.