Star Wars: The Last Jedi - 39 Things We Learned From The Director's Commentary
10. Snoke's Death, Kylo's Vulnerability
Adam Driver really wanted to hone in on what Kylo's thinking when he kills Snoke - he knows he can't be thinking actively, because Snoke will sense it, but he's looking for an opportunity to do something. His intent from the moment he walks into the scene is to betray Supreme Leader Snoke.
They then had to have Kylo be vulnerable in his 'join me' moment with Rey, with him making an emotional appeal rather than one based on power. The goal with Kylo was two-fold: to get inside his head and reveal that vulnerability through Rey's eyes, because we had to go along with her on that journey without just saying 'screw that guy, he killed Han Solo'.
But also, Johnson "wanted to bring him closer to being the villain that he wanted to be in The Force Awakens.” He's not Vader, but has come into his own as a sort of villain that the audience and Rey have an understanding of. The lightsaber was suspended exactly between them to show that they're equally matched.