No, I am certain that Arturia's standard hairstyle is not, in fact, a ponytail; I believe that generically calling it a bun would be more accurate than calling it a ponytail.
Seems also like more often Mordred has a straight ponytail, whilst Arturia keeps hers in whatever that bun is called.
And Arthuria's ponytail in black suit in Fate/Zero is lower ponytail, not upper ponytail like Mordred. Also, Casual cloth of Mordred is more "naughty" than pure "innocent" like Arthuria.