I have no clue about the Dancer class from FFXI. I never played FFXI, but have been playing FF games since the original Final Fantasy and the only "dancer" I know of is Mog from VI. To me, the woman in red looks like a Monk performing a kata.

ka·ta
ˈkädə/
noun
a system of individual training exercises for practitioners of karate and other martial arts.
an individual training exercise in karate and other matial arts.

After her kata, she then spars with the meditating male monk, using the exact moves that the male Monk is using.