Jaq'en gave Arya the coin, and that admitted her to the Temple of the Many-Faced God. He gave it to her in case she was ever out of options, and she was desperate. The Hound is dead, and the realm is hunting her. She had to leave and had no where else to go. So I am not sure much was on her mind except that maybe she would find Jaq'en there and get more help.
Her life was very similar to a faceless man prior to that anyway. Wearing different names, and identities to survive. She used it unskillfully, but did use Jaq'en to escape Harrenhal. So really, not much of a differentiation, except they gave her confidence, some training, and safety.
I don't think Arya ever really considered being a faceless man. At her arc in the Show she has what she needs to survive. She is no longer a lordling that only knows how to only order people around, but can survive in the streets, woods or as a kitchen worker or in some other capacity. I see her going straight for Walder Frey......next season. I think she gets on a shhip last episode this year.