Although swordplay is sometimes referred to as dancing, the term was also deliberately used to deceive others as to the nature of Arya's training.
As Ned Stark first confronts Arya with Needle in hand, he muses (from the book)
"Lyanna might have carried a sword, if my lord father had allowed it"
implying that it was not conventional for women of Westeros to learn sword fighting.
When Arya first meets the bravo Syrio, he states (from the book)
"You are late, boy"
further reinforcing the notion that girls are not taught to use weapons.
Syrio then explains the water dance (from the book)
"Remember, child, this is not the iron dance of Westeros we are
learning, the knight's dance, hacking and hammering, no. This is the
bravo's dance, the water dance, swift and sudden."
Later, here's the exchange between Sansa and Arya (from the book)
"How was your dancing?" "I'm sore all over," Arya reported happily,
proudly displaying a huge purple bruise on her leg. "You must be a
terrible dancer," Sansa said doubtfully.
implying that Sansa has no clue as to the actual nature of the "dancing lessons"
When Ned starts doubting Syrio's effectiveness and suggests alternate teachers, he again reinforces the notion that these lessons are "on the sly" by suggesting (from the book)
"Or I might have a quiet word with Ser Barristan."
As for why Jaqen H'gar understands, that's self explanatory when you know that he is also a bravo so one can assume he would be familiar with water dancing as a fighting form and the reference to it as "dancing".