[ by 'pet name', I mean a soft term like 'sweetie', or 'my human' or other things like that. I think they'd all have different ones which they'd prefer c': ]
Diavolo
Diavolo wanted to ask if you're okay and was trying to think of how he can help, but what actually comes out is a quiet "are you alright..." followed by an affectionate term of endearment that he'd only use for someone he cared deeply about. "Oop- pardon me," Diavolo gives you a bashful squint as he realises what he said, he's not sure how you'll take it and he's ready to apologise from the heart if he embarrassed you. If you say you found it sweet, Diavolo will give you a genuine smile and say he's glad and grateful that he could cheer you a bit with his oversight, but then he's also sincere as he tells you the reason he called you that is because you mean a great deal to him.
Barbatos
He bids you goodnight outside your door, before you both stare at each other in shock as Barbatos realises he just used quite a soft, silly pet name for you. The name itself is really only used in affection among demons in the Devildom, but you're familiar enough by now to understand it perfectly well. Barbatos immediately feels terrible for the slip-up, he hangs his head and admits he's grown quite fond of you but he still doesn't know what possessed him. Barbatos looks mildly comforted when he sees you truly like the name and you're not offended, and eventually he gives you a gentle smile, thanking you for being so kind when he's feeling out of sorts.
Simeon
He always feels comfortable and happy around you, so affection comes easily to him. When he accidentally uses a fond pet name instead of your real one, Simeon immediately smiles mischievously and looks all around you both, saying that he's surprised your demon companions didn't suddenly leap out of nowhere to prevent any further closeness. He'll beam at you even more if you laugh and shake your head at his dorky jokes, then he opens his arms for a hug and asks how you would feel about him maybe calling you a soft name like that in the future too.
Solomon
He'll laugh at himself and give you a regretful smirk, saying "ah, I'm sorry. That kind of came out of nowhere, huh?" Solomon looks a little more cagey as he tries to determine whether you might actually ever want him to be affectionate with you. When he can't read you well enough to be sure, he shrugs and offers "that being said, if you ever want to call me a pet name, you can. I'd quite like it, actually."
Lucifer
Lucifer's not exactly concerned about startling you with an unexpected show of affection, he just hates the fact that he caught himself off-guard even more than you. When you quietly ask him to repeat what he just said, thinking you must've drifted off in class and misheard him, Lucifer draws a breath, pauses, and then doesn't allow himself to look the slightest bit flustered as he slowly repeats what he said in front of your whole class: "would you please join me for an urgent student council meeting, my love?"
Mammon
"Ugh... dammit, I mean... no, I mean, I didn't mean-" Mammon gets tongue-tied trying to talk himself out of his own words, he's blushing so furiously that you immediately want to save him. If you offer a hushed "you should always call me that..." Mammon will just keep nodding and ranting saying "-so, ah, yeah, ya see now.. when I said that just now.. er, what I was actually thinking was-" then he stops dead silent as your response suddenly sinks in. He looks down at his feet as he quietly finishes with "...was that I wanted to call you that."
Leviathan
He happily blurts out the pet name he's been imagining saying to you (but only if you were both characters in an anime and his character was a lot more confident than real-Levi is right now!!) Then a moment later Levi processes that this has actually happened and he starts wailing at the top of his lungs, collapsing into you and burying his face in your shoulder. You find it sweet and a little bit sad that he trusts you enough to leap into your arms for a hug, but that he still clearly thinks you might hate what he said. Levi tries to pipe down as soon as he realises he's being really loud right next to your ear while he's clinging to you, and you give him a helpless smile, saying "noo, it's okay..." then you share what pet name you'd use for him too, so that he doesn't feel alone.
Satan
He'll attempt to keep talking and not acknowledge that he just used the most humiliatingly feelsy nickname for you he's ever heard, in the hopes that somehow you didn't notice. Eventually Satan can't stand it any longer and squints one of his eyes closed, like he's about to face up to something a lot harsher than just his feelings. "Just now... was what I said okay?" If you nod yes or express that you feel close to him too, Satan will duck his head and frown quite severely as colour warms his cheeks, then he murmurs "thanks. I guess I needed to hear that..."
Asmodeus
Looks at you in giggly anticipation, the fact that Asmo surprised himself as well as you puts him in even more suspense for your reaction. If you blush or say you don't mind his cute name for you, Asmo will happily throw his arms around your neck for a hug, jokingly exclaiming that now you've both opened the Pandora's box of goopiness and it will be adoring pet names all over the place from now on.
Beelzebub
"Oh. Sorry. I meant to say that in my head." Beel scrunches his nose and blinks at you as soon as he hears a gentle nickname for you come out of his mouth. When you're absolutely speechless, Beel asks if this is a happy surprise or a bad one. He'll pull you into a half-hug if you're happy, saying he's definitely going to say things like that again because it means a lot to him to see you smile like that. If you keep smiling he'll laugh and hug you tighter, saying you can stop any time now if you don't want to see him get all sappy and squish you with more hugs.
Belphegor
"Whoops. I might be really bad at this," Belphie ducks his head and gives a soft laugh, he can't help looking embarrassed as soon as he realises what he's said. If you ask what he means by that, Belphie hums and gives you a guilty smile, then admits: "being romantic. I guess I was planning on telling you another time, but whatever. I was going to ask if I could call you that sometimes, because that's how I feel about you. But it's okay if you say no." One of the things you find special about Belphegor is the fact that he truly does mean it, it won't bother him at all if you tell him you'd prefer your own name. Either way he'll just give you the biggest hug and say "alright. You got it."