That's just a part of what Moore said as well.
I just listened to the whole press conference for a second time. This is the complete flow of questions about Carr and Moore's answer. It's not the entire transcript, but it's an accurate summary.
Q: What he thought about Carr when viewing his tape and do you view him as the week 1 starter for the Saints with Moore as head coach?
Moore: Carr is a tremendous QB in this league. He's a starter in this league. He's a premiere player in this league. I'm real excited to team up with him and go through the process... (he went on to say he looks forward to going through that process with all of the players).
Several minutes later Triplet asked:
Q: Follow up on Derek, do you view that as having a decision to make at QB or do you view it as Derek Carr is the QB?
Moore: I view it as Derek Carr is a tremendous starting QB in this league. We're fortunate to have him and we're excited to go through this process, you know, as we build this roster together. Again, I meeting these guys for the first time in the last few days. We're excited to meet more of them and this will be a lot of fun.
I don't see anything at all "damning" that Moore said about Carr. One of the things I like about Moore is he seems more like a collaborator than a dictator. Even if Moore was deadset on Carr being the starter, I don't think he's going to say that in a press conference before he's actually had a chance to discuss it with Carr, because he knows Carr has a say in the matter. I think it's pretty clear that he would comfortable and confident in having Carr as his starter.
So that's the full context of the questions and answers about Carr that Underhill left out of his tweet that you quoted.