According to Smith's "Smallarms of the World", the first true bolt-action metallic cartridge rifle was the Swiss Vetterli. The Berdan and Needle gun's weren't quite there, though they certainly pointed the way. The Vetterli used a rimfire cartridge that was still sold as late as the thirties, due to popularity of the rifles. They were inspired by the American tube-magazine designs, but the Swiss wanted a non-lever design that was more useful for prone shooting. The American lever-actions were better for the mounted shooter, but difficult to use when prone.
Like I said, the features of the modern bolt action rifle did not come all at once. Dreyse was the first to make a bolt action rifle that used a cartridge. Your definition seems to be the first with a metallic cartridge. Someone else may define it as the first to use a metallic centerfire cartridge with a box magazine. The needle gun was the first to use a bolt and a cartridge and pioneered most of the features of the modern bolt action rifle.
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