The German word that they used was "Bund", which was the usual word for all kinds of alliances. The Swiss used the word for their alliance as well (next to the more famous Eidgenossenschaft), there was also a Schwäbischer Bund and a Rheinischer Städtebund in southern and south-western Germany. You could say it's a generic, descriptive word.

The oldest of the Three Leagues, the one of God's House, is quite often in sources referred to as "the people" or "the commoners of God's House", because they were the subjects of the bishop in Chur who formed that union in opposition to the bishop's plan to sell the territory to the Habsburgs, and in analogy to the Estates in other feudal polities. Their assembly was called the Bundstag "League diet". So in this case, "Bund" is really more of a description than a consciously chosen name.

Bund comes from binden "to tie", and the Latin liga from ligare means the same thing. In Latin documents about these polities, the word foedus is used too. This means "pact".

What is also interesting, is the fact itself that these were alliances of common people in a rural, mountaineous and overall rather poor territory. Other such Bünde like those in Germany were most often of cities.