I think that Logan was being honest with Shiv, when he said that he was going to give it to her. But at the end of the day, Logan doesn't want to leave the company yet. He wanted Kendall, but Kendall wanted to do things his own way, and Logan wouldn't stand for that. So... he opted for the next best option, which is Shiv. But as soon as Shiv got a whiff of power, she started doing the same thing that Kendall did... going AWOL. Logan doesn't mind naming a successor, if they're willing to just have his back and learn from him while he's still alive. This company means something to him, and he wants it run in the way that he'd run it. If Shiv could have just learned, like Kendall did, to respect Logan's expertise... I think he would have given it to her.
But she's acting like an entitled brat, who would rather run things than learn things... Logan recognizes that, and won't give his company to someone like that. Right now... Kendall is basically behaving exactly how Logan wants him to. If he can keep that going, I think Logan will reverse his decision and give it to Kendall. I don't think that he really cares too much about Kendall's betrayal... if anything, he probably admired it. It's something that he'd do, and he was always worried that Kendall didn't have the killer instinct.. which he now seems to have acquired.