Browning's started experimenting with gas operation in the 1889. The rifle operated off the lever, that he later used on the Colt 1895 Potato Digger was actually the second rifle he modified. The first one operated off of muzzle blast, though it would still be gas operated technically. Either way though, he was still a couple years behind Maxim.
The Mexican Mondraon was developed by a Mexican Army officer. It had some good features, like a 10rd box mag, etc. and some actually saw combat With German aircrew in WWI spotting planes before planes were normally armed. Small lots had been made in Switzerland.
However the first semi-auto rifle to see combat was the French St Etienne. The French started experiments in 1894 and eventually developed the St. Etienne. Gas-operated, rotating bolt, box mag, with the oprod running exposed down the side of the gun. Not to reliable with all the mechanicals exposed down the side, but it was the first to see combat, doing so in WWI.
1900 represents the year that many experimentals jumped on after the basic ground-breaking work had been done by Maxim, Von Mannlicher and Browning. After the true geniuses showed how it could be done, by 1900-1910 the Italians, Germans, Sweedes, Japanese, Danes, French, Mexicans, and several American compaines had semi-auto rifles. They were both recoil and gas operated, about half and half. Some worked and some were hidieous.