Hi, Sevens and guys,
After doing a fair amount of research and some connecting of dots, the .38 and .32 ACP story seems to go like this. Browning was a free-lance designer who sold his ideas to any gun company that would buy the patents or pay him royalties. But he worked and experimented mostly on his own, at least in the early days. That meant he had no source of made to order cartridges. So when he began working on his auto pistols, he took the most common revolver cartridges of the day, the .38 S&W and the .32 S&W and tried using them in his pistols. (His early patents clearly show rimmed revolver cartridges.)
But it was soon clear that rimmed rounds would not work well through a magazine. So he kept experimenting, making the rims smaller and smaller. Finally he got cartridges that worked and fed fairly well but, even though the rims were smaller, they were still there, and the cartridge case is supported (or "headspaced" if you prefer) on the rim. Browning called the cases "rimless", but we use the term "semi-rimmed." Browning used that idea in his .38 ACP, the .32 ACP, the .25 ACP, and the 9mm Browning, aka the 9mm Browning Long. The .38 Super is a heavily loaded .38 ACP. In all those guns the chambering has to allow for the rim.
Sometime around 1904, he either had a better idea or, more likely, became aware of the 9mm Luger, which is supported on the case mouth. The two cartridges he designed subsequently, the .380 ACP (aka 9mm Browning Short) and the .45 ACP, are straight rimless cases and are supported on the case mouth.
Jim