Tank - Tank hoped Neo was the one but we never really see him believe it. He seems ready to believe but just doesn't. Then Neo goes in, and does what no human has ever done. He fights the agents and wins. He gets everyone out alive. The moment he pulls Trinity from the helicopter, it becomes apparent that he achieved his objectives. He Won (or they won). He fought agents and won. This "victory" gives Tank the "He must be the one moment".
Trinity - She is going to fall in love with the One. That's her part. So she gets emotionally yanked around, half falling in love, half not wanting to, because it means loving a doomed man. So she believes he might be the One, as she is starting to fall for him, but is also hoping he isn't because that would mean she could love a normal man, instead of some savior figure. Then, she has to face the fact that she didn't let him come alone, and that, in the heat of the moment she trusted him as instinct. She is smacked in the face with the truth that she trusted him (a essential part of love), that he risked his life for hers, and that the trust was rewarded. Her "revelation" was less about the one, and more about a man she was starting to love. Even though that love may mean that he was the One.
Morpheus Morpheus believed he was the One. But there is a long gap between believing and sitting there talking with your "Jesus". Not only that, but his moment was way more "I F'ing told you! See! All you haters doubted me, but he IS the one, so suck on that!" Of course Morpheus has too much class to say it like that, but.... That's the moment anyways.
Neo Neo still doesn't believe. He is starting to, but he is still hesitant. His "suspension of disbelief" doesn't happen until after he "dies". He just see's it as a kind of "Of course it worked. Why wouldn't it. I didn't do anything you didn't show me in the simulator." Neo didn't want to be the One. The whole reason for going back is because he didn't want Morpheus to die for a fake "One". He felt guilt because Morpheus was making the ultimate sacrifice for him, and he wasn't the One.
So what really happens is that Neo, does something that no other human can do by rescuing Morpheus from agents. The three with experience realize this now are put deeper in the "he is the One" camp. The noob stands there going "But all I did was press Q" and while he gets a confidence boost, is not swayed.
As stated in the comments, I drew on the fact that I recognized a strong relation between "The one" and Jesus, and that is why I think that Trinity is fighting her urge to fall in love with Neo. It's worth pointing out that no matter how strong the analogy is/was "in universe" there was no Jesus, and I can not come up with any reason that "in universe" Trinity would think Neo "doomed". I still think she does, and I still think it falls under all savior figures are doomed, but there is no "in universe" explanation for that, as far as I know. (Except that maybe she knew he had to "return to the source" and that it would likely mean death, but they didn't get into that in the first movie.)