Batman questions anyone and anything and has a plan for any eventuality, PARTICULARY those that pose a global scale threat. It’s nothing personal, it’s just reasonable precaution. That’s basically his true superpower. He also does trust Superman as a person, as a colleague and friend. I don’t think he ever considers there to be a true risk that Superman turns on humanity of his own will. However, Superman is susceptible to mind control, to magic, to unpredictable forms of kryptonite. And he is not the only living Kryptonian in existence either. It would be stupid not to plan for such threats.

Lex depending on the version, may or may not think that Superman actually poses a willful threat to humanity. But even if he also trusts that Superman is what he appears to be, a selfless hero that only wants to help people, he probably hates that idea even more. He usually doesn’t distrust Superman’s intent. He hates what it says about and does to human-kind, and by extension, himself. He things depending on an alien demigod will make humanity weak and complacent. He thinks that Superman holds the Earth back from reaching their potential. That it permanently neuters them from become Supermen themselves. So he makes it his mission to ruin Superman however he can. If he can kill him, good. Not a problem anymore. If he can publically discredit him, sow distrust across the globe, that’s good too, maybe better. People who distrust him won’t depend on him and may, in fact, fear him. As a result they are more likely to better themselves, their technology, their science, to rival and fight back against Superman.

TL:DR: Batman takes precautions. Lex hates and attempts to kill or sabotage. They’re not the same.