The second link you posted uses a set of spelling conventions known as Taraškievica, named after the man who codified it, Branisłaŭ Taraškievič. His work was published in 1918, and more Belorussian language reforms were planned. However, the Soviets objected to this sign of Belorussian nationalism, so they instituted a new set of reforms in 1933, which brought Belorussian orthography more in line with Russian. The Russian authorities went so far as to arrest those using Taraškievica. Source

In recent years, beginning with the perestroika of the 1980s, there has been a movement to revive Taraškievica. So, that's why there are two Belorussian Wikipedias!

My knowledge of the context of Belarus-USSR relations during that period is limited, so I would love to hear some of the context of these political struggles.