Ah, the Spock poster didn't load for me the first time I clicked through to the story, so I though they'd had to change Spock's name (presumably, the studio clearances department found exactly one person in Wyoming named S'Chn T'Gai Spock, and they had to change it, just like what happened with Jackson Archer), but it looks like it's just an autocorrect bug that changed it to "T'Gain." Which I can sympathize with on a very direct level, having "Gian" in my name, which my phone also likes to change to "Gain."
Anyway, looks like the fanfic crew won out, picking that one instead of D.C. Fontana's tongue-in-cheek "Xtmprsqzntwlfd."
I'm amused that this, from a certain point of view, vindicates the post-ST09 trend in fanworks to refer to the character as "S'Chn T'Gai Spock" ubiquitously, more regularly than several canon names and facts are drawn upon. While the "S'Chn T'Gai" family name is official, but non-canon, originating in Barbara Hambly's Star Trek/Here Come the Brides crossover novel Ishmael, an equal degree of deference in contemporary fandom could've been afforded to "Xtmprsqzntwlfd," a tongue-in-cheek suggestion made in response to fan-letter by Dorothy "D.C." Fontana, a writer on the original Star Trek television series who made several important contributions to the character of Spock and his Vulcan background, which had historically also found some degree of purchase within the fandom (though never really winning out over the simplest answer, that his name is just "Spock," which of course is inaccurate, as the existence of a second name had been established in the episodes "This Side of Paradise" and "Journey to Babel").
However, as in the case of "Hikaru Sulu" and "Nyota Uhura," the long-delayed establishment of a canon character name reflects the consensus choice of the fandom from among several officially licensed and behind-the-scenes options and suggestions. In this light, it's unsurprising that Strange New Worlds, in canonically establishing a second name for Spock, would choose the more popular option rather than an (admittedly joking) one of arguably superior provenance, or even inventing a new name. Given that fan creators are frequently looked-down-up or treated as pariahs in officially sanctioned media venues for both legal and social reasons, I find this concordance of this piece of "fanon," (which is particularly popular among fans of a generation which could not have been exposed to Ishmael, a moderately obscure tie-in novel, directly) and "canon," amusing and worthy of a small, winking acknowledgement.