That panel stood out to me too. But it also reminded me of Yukio in a different way. Not the flaming bit, but the shape of the eyes/eyelids. Here I'm just talking about basic inheritance, not demon-y stuff. Like, 'you have your father's nose'. Remember Shura said Rin has Yuri's eyes. Yuri's eyes were more pointed and up-turned and look like Rin's. Yukio got his eyes from his human father (005) then.
Oh, yeah, that's pretty "certain", I guess. Yukio's eyes' shape totally DOES look like Goro's. Just like Rin's totally looks like Yuri's (I couldn't find better panels, ugh):

About Yuri, since we mentioned her, we still don't know much about her but we know she's "related" to Asylum/S13. Someone theorized she could be a clone too (and even a clone of Egyn 'cause of the similarity of their names) but reading the BTS arc (and, well, maybe it was easy to guess even back in chapter 85-86) I think there's proof she wasn't a clone but, I guess, just an orphan. When Lightning and Suguro were consulting the clones' lists, Yuri's name was not found in them. Lightning knows her name (he stated he read reports of the Blue Night and mentioned Yuri's name).
And, in the BTS arc, after Yukio reads the same files, he comments "I couldn't find anything [...] about Yuri Egin." Now, unless Yuri was given this name later and in the files she had a different one (I doubt this since Shiro's name didn't change, but maybe it's possible) the only other explanation I can think of is maybe Lightning DID find Yuri in those lists but he didn't put the file in the usb he gave Yukio. But I can't think of any reason for this, so. Anyway, just a random comment.
Which makes me wonder... Telepathy between demons is supposed to be a demons-only thing, right? That's why Rin can understand Kuro, or sense demons in general, and Yukio not. To be honest, I had a doubt in chapter 93 that prolly Yukio could hear Satan's voice but not understand his "language". Such as Kuro's meow which Rin understands in "human" language. I mean, Yukio can hear Kuro's meow but only Rin can understand what Kuro's "saying".
But it looks like Yukio CAN understand what Satan says. At least, from what we can get by Satan commenting about Shiemi in chapter 97 and Yukio remarking. Sooo... Yukio can understand a demon. Okay, is it a new ability he unlocked from chapter 93 too/it only works with Satan 'cause he literally lives "inside his eyes"/is it connected to a possible "blood relationship" between them which, after all, they share (even if tinier compared to Satan/Rin's)/Satan is actually "possessing" Yukio's mind (at least a part of it) too (maybe through his eyes) and that's why he can "manipulate" it like he sometimes did in old chapters (Yukio's blackouts/walking to places without realizing, and so on...)/Yukio is actually a half demon, after all?
On second thought, after reconsidering this part, I think I made confusion. I talked about telepathy but maybe this is not the "way to communicate" Satan is actually using. Humans CAN understand demons when they talk (well, I can think of Uke, Mike, the Kraken, Hachirotaro, etc.), they just can't "read the thoughts in their minds"/"communicate through their minds".
So the point is: is Satan communicating with the twins through "talking" or "telepathy"? I think this should be clarified first.
My guess is Yukio had (or still has) the potential to be a full half-demon like Rin. Like he got the basic inheritance or genes that link back to Satan, but didn't actually get the demon heart/flames. The process was stopped or failed partway because he was weak/and or something was done to him. Remember Shiro could have just said originally in ch1 to Rin when he asked about Yukio, that there were never any flames involved with Yukio to begin with.
Instead he says his weak body couldn't take them or whatever, so he remained human and didn't inherit them. That implied there were flames there for Yukio at first, but failed to get to him. That's why I think it's both a yes an no that Yukio inherited the flames. His body has the inheritance factor, so he's got the basic blueprint for it, but he didn't inherit it since the flames never came all the way. It could be permanent damage and that it was a one chance to get it right at birth. That would mean Yukio would always have this link to Satan only in the eye(s). Or it's not permanent and there's a way to correct it with Lucifer's methods. That's something at this point we need to wait and see.
*screams* I like this. And it made me think/remember of other things. I mentioned this theory before but I want to try and connect it with what you said and try to make up a new. I theorized before that Yukio's "Give me your flame." (at least based on Viz's translation) plus his general wish for power/strength and his "obsession" to be stronger than Rin could depend by the fact Yukio's own flame power was somehow "absorbed" by Rin while they were in the womb ('cause he was physically stronger), leaving Yukio with a sort of "unconscious" crave for that power he was separated from at the moment of his conception. Like, Yukio feels like something is "missing", his own power, and unconsciously craves to get it back. (and this is why he's so fixated on this, it's like an instinct)
But, considering we now know Mephisto could've been training Rin specifically to make him capable to host a huge flame power (Satan's) what if Yukio's "flame power" (which, as you said, was apparently "there for Yukio at first, but failed to get to him") was instead -purposely- transferred to only one twin (Rin, 'cause he was physically stronger) so he could be trained to endure more flames and develope a stronger "tolerance"/"resistance" to a larger amount (Satan's)?
Another one of my headcanons is that Rin possesses innatural strength (and he was able to "manifest" it even before he was "awaken" to his demonic form), while Yukio not, 'cause Rin may have inherited it by the enhanced genes of his human father. Now, I don't know if by "enhanced" Mephisto meant "a more resistant" body (to physical damages) or a body with a higher ability to regenerate (heal itself) or simply a human body capable to better armonize with demon genes or all of these. But Mephisto also added "strongest", the strongest enhanced human. Strong, as in "physically" strong?
Anyway, even if it's a combination of all these things, the result was that prolly Rin inherited a "strong enhanced" body by his human father genes (and was "powered up" by the demonic ones of Satan). While Yukio prolly didn't got those genes (or he would've manifested it somehow, either by being able to lift heavy objects, run faster, heal faster, and so on... But it looks, and the manga proved us countless time, Yukio can't do any of this (heck, his arm's been broken for 17 chapters now/a couple months in manga-time)) so, Yukio may have got a "weaker" body 'cause he didn't inherit the "enhanced" genes.
Or, and here it gets tricky, someone experimented on him and manipulated his genes on purpose to "weaken" him and, consequentially, "separate" him/making him unable to sustain his share of flame power and make it as Rin ended up being the only twin to inherit both flame powers.
Now, as you said, I wonder if by "unlocking your potential" Lucifer actually meant "doing something" to undo what 'caused Yukio to not gain access to his power and he may offer some "potion"/medical procedure to make him "unblock" his enhanced genes which would make him capable to host the flame power again.
Satan was pleased with Samael holding up his end of the bargain,
Which makes me wonder, will we now find out what was -Satan-'s end of the bargain? Maybe they're teaming up against Lucifer (since I hope this is really Mephisto's plan as he declared)?
I mean, if my pet theory that Satan has protected Yukio from getting possessed by any other demon is true,
Whhhhich makes me wonder, now that Satan got what he long wanted (break the Koma sword) what's left for him to do with Yukio? There's more he wants to do with Yukio or he may now decide to let him go or... he just can't (for now)?
I think that might be an evidence that at least part of the flame is Yukio's, not all belongs to Satan. His right eye still stay round shape while the left eye became the narrow demon form. Looks like it's to emphasize his right eye is all human --- namely it's 100% Yukio's. So the flame in his right eye probably is Yukio's flame.
Yup, this too. I understand Japanese's plurals are kinda "confusing" so there's this "eye/eyes" which is not that clear, but the fact remains that Yukio's eyes do BOTH change shape when they "activate". One is more Satan-shaped, okay, but his right one does at least turn blue/his iris change into "cat eye"-shape (similar to Rin's when he's in demonic form)/it does emit some lil flames. When Yukio covered his left eye in chapter 97 Satan said something like "don't cover me up" so it looks like Satan can only see/he's only "inside" Yukio's left eye or else he could still keep watching from Yukio's right eye. Then what about his right eye? Intriguing.
Edit: very cracky theory. What if the flames Rin can use while the sword is sheathed and the ones he can use while his sword is unsheathed are two different flames? And one of those are Yukio's?