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Rey does not attempt to resist the dark side of the Force during her training with Luke Skywalker. Luke tells her to resist and she fails and he is clearly outraged. He says "It didn't scare me enough then. It does now.", referring to Kylo Ren.

Luke appears very disturbed by this and storms off. When seeing this, it appeared clear to me that Luke would cease training her or they would have some sort of an impasse, but the very next scene between them starts off with Luke saying "Lesson 2..." and they move on like nothing happened!

Why does Luke continue to train Rey despite being scared of her and what she might become?

Luke was never trying to teach Rey how to use the Force. He was trying to show her why the Jedi needed to end- and, by extension, why he would not train her.

The first lesson was supposed to show Rey what the Force actually was; she already knew how to access it, but she had no idea what it was or how dangerous it could be, like a child playing with a loaded gun. It made sense to show her that she was only a small part of the Force, and how she could not control it herself.

The second lesson was about how hubris destroyed the Jedi and created the Empire. So again, Luke is trying to show Rey that the Force is dangerous, and how she should definitely not rely on it to find her parents or save the Resistance.

The way I see it, Luke couldn't just kill Rey: last time he tried that it backfired horribly (and at this point he has neither lightsaber nor the Force to help him). He also couldn't train her to use her power, because again that didn't go so well with Ben Solo, and Rey looked like she was going down the same path. Instead, he tried to show her that she didn't own the Force, and could never master it to get what she wanted, and if she tried it would only end in tragedy for her. Essentially, as Yoda explained, he taught her about failure, so she could grow beyond it.

This actually isn't how it seems. That "lesson two" is the precise logical conclusion of Luke's fear of Ben's - and now Rey's possible - failure to the Dark Side.

The Novelization covers the exact sequence:

• First, Rey has that first lesson when she is drawn by the Dark side.



• 
  

  Then, Luke observes her practicing with the lightsaber - NOT as part of instruction, but because she decided to do it herself. They have not spoken at that point, since the first lesson, yet.

  
  
  

  
  

  Rey stood opposite an outcropping of rock, practicing sparring with her staff.
  
  On Jakku she’d rarely neglected such practice—she needed to defend herself against threats ranging from marauding Teedos to fellow scavengers willing to kill for a valuable bit of salvage. She’d let things slide since arriving on the island, though, and now she wished she hadn’t.
  
  She was rusty, for one thing. But the exertion also helped blow away the fog and frustration that had enveloped her. Here there were no gnomic Jedi utterances or malevolent visitations to contend with—just the need to keep her staff spinning and jabbing.
  
  Rey didn’t allow herself a breather until she was sweating freely and her arms and shoulders ached. Leaning on the staff, ignoring the curious porgs circling overhead, she spotted Luke’s lightsaber peeking out of her bag.
  Should she?
  Of course she should.
  (Chapter 17)

  
  

  
  
  

  Note that she very explicitly starts her excercise to contrast with Luke's weird "teachings", and deliberately decides to practice with Lightsaber next.

  
  


• 
  

  Then, as Luke observes her destroy the rock outcropping (and the Caretaker's cart), Luke starts talking to her - at her prompting, mind:

  
  
  

  
  

  The suns were dipping toward the horizon as Rey and Luke entered the Jedi temple, facing each other across the font in the center of the ancient space.
  
  “So,” Rey said.
  
  “So.”
  
  She shook her head. “Nope, you start this time.”
  
  “I’ve shown you that you don’t need the Jedi to use the Force,” Luke said. “So why do you need the Jedi Order?”

  
  

  
  
  

  As you can see:

  
  
  
  
  

  • He didn't want to talk to her until he saw her practicing with the Lightsaber (at which point he knows that she would risk turning to the Dark Side with or without his instruction).

  
  

  • Even then she had to prod him into the conversation.

  
  
  
  

I Just re-watched The Last Jedi and this is the impression I got:

Luke knows that he trained Kylo Ren, but kept knowledge of the Dark Side hidden, or at least downplayed. This allowed Kylo to be seduced.

Luke realized that withholding knowledge is dangerous, and rather than let another powerful force sensitive out into the world, he continues to train Rey. He stresses his thoughts on why the Jedi, specifically, needs to end. We don't get to see a long diatribe about this, but my brain just assumes it happened.

I suspect it's going to lead into a canonical creation of Grey Jedi, Each individual containing within themselves balance in the force, as opposed to the forced duality of Sith and Jedi.

This is all just speculation on my part.