In the Andante Fundamentals, Greg addresses not getting a sympathetic oscillation. Don’t worry about it. The takeaway is: do not pinch the middle of the lips and do not overblow. As for the lips vibrating after pulling the mouthpiece out, this may be caused by various factors. I have had this experience, and I am not sure why it is so. I hypothesize that there might be a change in mouthpiece pressure on the lips while I’m removing the mouthpiece from the leadpipe. My advice is don’t worry about it. Understand the process and sensation. Releasing air through the leadpipe is what playing should feel like, no kick nor pinch. A little more pressure, a little more aperture corner tension, a change in the mouthpiece’s angle, perhaps a change in the jaw or tongue position may cause the mouthpiece buzz. Continue to experiment and maybe you will figure it out, or not. You are learning what playing the trumpet should feel like.
One of the amazing things is that playing the trumpet is a lot like Newton’s third law of motion. A bat hits a ball and the ball hits the bat at the same time. You play the trumpet and the trumpet plays you.