I stopped watching Game of Thrones ages ago, but I just watched the clip in the OP and it made perfect sense to me.
Arya likes her little Needle sword, and knows a thing or two about sword fighting. She asks Brienne (who is clearly a badass) to teach her more about sword fighting. Brienne is like "LOL no, go away little girl." So Arya hits her with some loyalty pledge thing. Brienne is like "Damn, you got me. Okay, I guess I have to teach you." Then Arya pulls out Needle, and Brienne tells her to get a real sword, but no, Arya wants to do it this way, so... fine. Let's get this over with.
Brienne does one simple chop, and Arya uses a combination of her quickness and some deflections with Needle to dodge the chop and get into a striking position. Arya acts smug about it, so Brienne ups the difficulty and throws a combination of attacks at Arya, which Arya successfully dodges, and responds with a combination of her own, light and quick enough to score a point on Brienne's hand.
Now Brienne's taking the "fight" a lot more seriously, puts forward a better series of attacks which Arya responds to as if she was an actual fighter, scores another small point against Brienne's leg, and Brienne gives Arya a kick to increase the distance and remove Arya's closeness advantage. Brienne immediately has a reaction of "Oh shit, did I go too far against this little girl?" But then Arya gets up with a cool flip and she wants to continue. Brienne almost seems to be smiling, because she's impressed that Arya seems to be an actual fighter, worth teaching.
Brienne unleashes a dominant series of attacks that result in Needle being knocked out of Arya's hand, and rather than yielding, Arya draws her knife. But Brienne saw that coming and grabs Arya's wrist, but Arya flipped the knife to her other hand, and now both of them are in striking positions (basically, it's a tie). Both of them are smiling. Brienne is impressed, and Arya is pleased that she managed to impress Brienne.
I don't think there's anything in there that says that Arya "defeated" Brienne, or anything that suggests Brienne was going all-out. Brienne basically won the fight when she kicked Arya, and clearly didn't want to do that, and I think the main takeaway was that Brienne was impressed by Arya's general competence as well as her guts and trickery.