Some folks don't know how Browning worked. He was a "free lance" designer who developed his designs, usually made working models, then sold the idea to whichever gun company gave him the best deal. His earlier Winchester designs were sold outright to the company, for a lump sum. Later, on his autoloading shotgun, he tried to work out a royalty deal, where he would receive a certain amount for each gun made. Winchester refused, which resulted in the A-5 being produced first by FN, then later by Remington and Savage, after Browning made separate deals with those companies.
On the pocket pistols, Browning sold his first design to FN in a royalty arrangement, which made him a wealthy man. Then, around 1902, FN wanted a more powerful pistol to compete with the German Luger, but didn't care for the Colt design. They asked Browning to come up with a new pistol that would have good power but be simple and easy to manufacture. The result was the FN Model 1903 in 9mm Browning Long, essentially a shortened .38 ACP. It was fairly successful, but no world beater.
Browning interested Colt in the design, but Colt felt that the current dual-link guns were adequate for the holster pistol market and asked for a pocket pistol. Browning took the FN M1903 design and scaled it down, making the Colt M1903 in .32 ACP, using the same cartridge as in the FN M1900, called the 7.65mm Browning in Europe.
Later, there was a customer demand for a more powerful pocket pistol. So Browning and Colt worked to modify the M1903 hammerless pocket pistol to fire a new cartridge, the .380 ACP. The added "0" distinguished the shorter round from the older .38 ACP. Since by this time, Browning had learned about Luger's idea of supporting (headspacing) the cartridge on the case mouth, the new cartridge was rimless and untapered. The .380 pistol, called the M1908 by collectors, came out in that year.
Meantime, FN was experiencing lagging sales of the Model 1900 which was becoming outdated, and looked it. So they asked Browning to produce a new pocket model for them. Using pretty much a different system, Browning came up with the Model 1910, available in both the 7.65 (.32 ACP) and the .380 ACP, called the 9mm Browning Short in Europe. Later, in response to military requests, the grip was extended and the barrel lengthened to produce the Model 1922. The slide was lengthened by adding an extension in place of the bushing.
Sometime in those years, Colt and FN made an agreement to not tread on each other's toes. They divided the world into three areas: in North and South America, Colt would reign, and FN would not sell their guns; in Europe and the Middle East, Colt would give way to FN; most of Asia and Africa, both companies would sell in competition. That agreement lasted until after WWII, explaining why FN-made guns were very scarce in the U.S. until the 1950's, except for WWII GI "bringbacks".
Jim