Did Luke Skywalker Actually Turn to the Dark Side when he Defeated Vader and Chopped off Darth Vader’s Hand?

That depends on what 'turning to the dark side' means. From what we can tell, many characters have drawn upon the dark side for temporary uses, but from what we see, such actions do not inherently 'turn' someone to the dark side.

Turning the Dark Side: A Willful Decision

Turning to the dark side seems to entail a conscious decision of the will to ally oneself with the dark side and surrender to it. For example, in the novelization of "Revenge of the Sith," as Anakin kneels before Palpatine, Palpatine asks him, as part of the Sith initiation, 'It is your will to join your destiny to the Sith Lords forever?' Anakin's reply, 'yes,' marks his true 'turn' to the dark side, and he receives his new Sith name.

However, it's important to note that Anakin had already committed many very dark actions before this point: he killed every man, woman, and child in a Sand People village to avenge his mother, force-choked Poggle the Lesser to interrogate him, dismembered and impaled Admiral Trench, beheaded Count Dooku after disarming him, and cut off Mace Windu's hand to protect Palpatine, among other things. Despite these actions, which certainly demonstrate significant grievousness, Anakin still felt remorse and guilt for his actions. It was only when he faced the choice of doing the right thing or the selfish thing that he made that fateful volitional choice to embrace the dark.

Luke Skywalker's Dilemma: The Dark or the Light?

Fast forward to Luke Skywalker's confrontation with Darth Vader on the Death Star II. Luke unequivocally drew upon anger, hatred, and fear when he attacked his father, as Vader threatened Leia. He continued to press the attack, seemingly intent on finishing Vader—until the moment Luke cut off his father's right hand.

This scene so clearly echoes Anakin's confrontation with Count Dooku on the Invisible Hand and Anakin's choice with Mace Windu in Palpatine's office. Luke then stands there, looking at his father's stump with the wires from his artificial limb hanging out, and looks at his own artificial right hand. He undoubtedly remembers his vision on Dagobah in the cave: if he kills Vader in cold blood, he becomes just like him.

This is Luke's Jedi trial. It is also where he shows he truly is of the Light. He does what the Jedi of Anakin's day should have done and refuses to fight. He throws away his lightsaber and announces, 'I am a Jedi.'

The Clear Difference: One Who Chooses and One Who Refuses

This scene shows clearly the difference between the father, who chose willingly to embrace the dark after opening himself to it, and the son, who opened himself to it and refused its offer.

It turns out Yoda was wrong after all—once you start down the dark path, it does not HAVE to dominate your destiny. You can choose. Luke chose the Light.

Hope and Redemption: Anakin's Decision on Dagobah

And in possibly my absolute favorite scene in all of Star Wars, unknown yet to us as the audience, the Anakin inside Darth Vader also chose the Light once again. The story doesn't come right out and say it, but I admit this is head canon, but also very plausible given the parallels with Anakin's own experiences—the moment Luke refuses to kill him and refuses the dark, I believe Anakin begins to turn back to the Light. Luke has shown him the way he did what Anakin did not do with Dooku and showed that someone who has embraced the dark had a choice still to turn back. Anakin sees in that moment, contrary to his earlier statement, that it isn't too late for him. He can choose, and he does. He, of all people, becomes living proof that Yoda had failed to fully comprehend: 'Love can ignite the stars.'