Actually, Louis Armstrong played correctly; the first thing a brass instrument player learns is “do NOT puff your cheeks”. Dizzy Gillespie was probably self taught and young and no one told him not to do it.
He did prove that it does not hurt your playing to puff out your cheeks. :-)
I play a Bach Strad M37 with a 1 1/4C mouthpiece. I do not puff my cheeks because I do not press hard against the teeth like he must. I learned low lip pressure playing from Keith Clark, of Houghton College and formerly first chair US Army Band in Wash., D.C. He in turn was a student of Armando Ghitalla who perfected the technique. I studied under him every day for two weeks for each of two summers at the ‘74 and’75 Csehy Summer School of Music, Muncy PA.
Seargeant Clark is famous for cracking a note when he played taps for John F. Kennedy’s funeral. They said in the press that he was emotional and that he was so touched and moved, so that the people in turn took the cracked note as a symbol of our broken hearts. Awwwww...but no. Truth is he told me he had been standing in the cooooolllld for hours and then tried to play with no warmup. In my opinion, this technique is too fragile to be practical for real working trumpeters except for Ghitalla and Voisin maybe.
Dizzy Gillespie has double laryngoceles which is extremely rare. In addition, he has a condition in his cheeks called “glassblowers’ disease”.