Wikipedia lists “White, Orange & Green” two times, both given as part of a set that also includes “Tabhair dom do Lámh” («White, Orange & Green, followed by Tabhair dom do Lámh» and «White, Orange & Green & Tabhair dom do Lámh»). The Session lists “White, Green, And Orange” and “White, Orange, And Green” as aliases for the work and the discussion related to it also only suggests that it’s a song sung in a set with Tabhair dom do lámh, though maybe it uses the same melody?
Either way, I would not consider the two the same work. Tabhair dom do lámh is an instrumental piece originally written for harp. If someone(s) added words to it later, that work with lyrics should be its own work, not combined with the instrumental original. Tabhair dom do lámh is at least from 1603, and the notion of “white, orange, and green” for the Irish is from the 19th century, a couple hundred years after the composition of the tune.
Looking up “White, Orange, and Green” on mudcat.org, I found this thread:
Which says that “WO&G” is indeed words set to the tune of Tabhair dom do lámh, so I maintain that they should be two separate Works, although they should be linked.