Around 1901, several countries were interested in FN pistols, but wanted a better pistol than the Model 1900 and a more powerful round than the 7.65mm Browning (.32 ACP). FN did not make a locked breech pistol at that point (the BHP was over three decades in the future), so Browning decided to get as much power as he could out of the blowback design he was already working on. He did that with the FN Model 1903, using a cut down .38 ACP case to get a cartridge that would be safe in a blowback pistol but still have a reasonable power level for miltary use. The 9mm Browning Short (.380 ACP)had not yet appeared, and the cartridge was first called simply the 9mm FN or 9mm Browning. The "Long" was added after the "Short" appeared.
There is some question as to which Model 1903 came first, or which was a "copy" of the other. The FN patent was issued first (1901), the Colt patent not until 1903, but Browning was clearly working on both at the same time. My personal thinking is that the FN was seen as the original and more profitable (the word "profit" was never far from JMB's thoughts) because it was intended for sale in large quantities as a military pistol. Unfortunately, by the time it came out, other pistols, of greater power, had come on the market and the Browning was bought in quantity only by Sweden; the smaller Colt version eventually was produced in much larger quantities for the civilian market.
Jim