
So, the lore says: "In colloquial speech amongst close friends and companions, sometimes the tribe letter will be dropped from the first name." On that basis, those who know about the traditional culture of Seekers of the Sun might assume -- lacking other context -- that a dropping of the tribe letter indicates informality and closeness. However, from the word "sometimes," we know that this isn't a universal rule. Each miqo'te is going to have his or her own interpretation of when that's appropriate, if ever. Y'shtola clearly doesn't go in for it even among her close friends, whereas F'lhammin does with her daughter. There's definitely personality differences between the two of them which may account for their choices. From a "prescriptive" lore standpoint, I don't think we have enough information to say how common the devs intended prefix-dropping to be other than "it happens." 
As for actual character examples, L'yhta (raised tribal, but now quite proudly Lominsan) doesn't typically drop her prefix, though she's getting close enough now to a few people to be willing to do so. In her tribe, only immediate family members would drop the prefix, and then only in private. Despite no longer being even remotely affiliated with her tribe, she still holds onto this particular bit of deep enculturation. However, her nephew who's followed her back to Limsa Lominsa, L'ehli, drops his prefix whenever he's friends with someone as a show of solidarity and as a sort of way of thumbing his nose at his tribe. Another miqo'te she knows, R'kyala (raised tribal and with a more positive view of it than L'yhta), rarely uses her prefix and goes by Kyala with just about everyone. K'lyhia and L'asah, two other miqo'te in the Ivory Tower, seem to never drop their prefixes and certainly not in public.
Among player characters, there seems to be a lot of variance in behavior, so I agree that, given the lore and the way players interpret it, this is one of those situations where individual preference is key. There's no real right or wrong in when, if ever, a character chooses to drop their prefix. That said, I think the lore makes it clear that it's reasonable to assume -- without other context or knowledge of that Seeker's personality -- that when a miqo'te drops his or her tribe prefix, it's a sign of informality and closeness. Taking the example of R'kyala above, anyone who knows her knows that she doesn't see it that way at all (except for maybe the "informality" part).

As for actual character examples, L'yhta (raised tribal, but now quite proudly Lominsan) doesn't typically drop her prefix, though she's getting close enough now to a few people to be willing to do so. In her tribe, only immediate family members would drop the prefix, and then only in private. Despite no longer being even remotely affiliated with her tribe, she still holds onto this particular bit of deep enculturation. However, her nephew who's followed her back to Limsa Lominsa, L'ehli, drops his prefix whenever he's friends with someone as a show of solidarity and as a sort of way of thumbing his nose at his tribe. Another miqo'te she knows, R'kyala (raised tribal and with a more positive view of it than L'yhta), rarely uses her prefix and goes by Kyala with just about everyone. K'lyhia and L'asah, two other miqo'te in the Ivory Tower, seem to never drop their prefixes and certainly not in public.
Among player characters, there seems to be a lot of variance in behavior, so I agree that, given the lore and the way players interpret it, this is one of those situations where individual preference is key. There's no real right or wrong in when, if ever, a character chooses to drop their prefix. That said, I think the lore makes it clear that it's reasonable to assume -- without other context or knowledge of that Seeker's personality -- that when a miqo'te drops his or her tribe prefix, it's a sign of informality and closeness. Taking the example of R'kyala above, anyone who knows her knows that she doesn't see it that way at all (except for maybe the "informality" part).

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Quality RP at low, low prices!
((about me | about L'yhta Mahre | L'yhta's desk | about Mysterium, the Ivory Tower: a heavy RP society of mages))