Yuri's phone does not have many saved contacts. He doesn't have many people to save numbers for. Since he moved to St Petersburg on his own at 8 years old to focus on training, he was pretty much by himself. Having not made many friends back in Moscow, he simply stopped trying here. There was skating to focus on, and earning rent the rest of the time. Of course, there are child labor laws, but also of course, there are people willing to pay a desperate child to dishes in the back of a kitchen. Yuri didn't have time to spend with people outside of school, so he didn't bother to strike up any friendships, knowing they wouldn't mean much anyway.
Most of the contacts on his phone are just the necessaries. Coach Yakov and the whole team, the ballet Madams and a few fellow ballet dancers, his costume designer, the landlord, maintenance, some of his past employers, and a few skaters he's met a competitions from here and there. Of course, he has his grandfather, who somehow still manages to use a landline. So his text history makes for pretty dry reading, just mostly business communications.
He does get the occasional text from Mila and Georgi, typically funny things that's happened at the rink, usually about the other. Or Yakov's reaction to Viktor's antics. Mila sometimes gossips about the pair skaters, they're-so-in-love-when-are-they-going-to-marry that kind of thing, and also teases Yuri with pictures of cat-themed clothes, accessories and objects. Usually it's just Hello Kitty. Despite being less than a good conversationalist in replies, he does appreciate these texts. The apartment gets quiet, what with it being him alone with Pushok in that tiny place, so the texts give his ringtone some good use and fills the room up somewhat. He absolutely loves "Welcome to the Madness" and doesn't get to hear it often enough, though that opening riff keeps startling Pushok. One would think the cat would have learned by now.
Now the effects of this prolonged solitariness and focused competition births within the man a guardedness and suspicion of others' motives. In the absence of care, the harsh indifferent ways of world breeds a defensive, almost feral flinching from care that is too freely given. Mila and Georgi took years to warm up to Yuri (or rather vice versa), so when Yuuko first texts to ask how Yuri is doing, he does not reply, suspecting her of snooping of Yuuri's behalf. She sends a follow up, "You must be so busy training hard for you senior debut. Good luck! We're cheering for you!"
Yuri does not know what to make of it when Yuuko texts again when Yuuri has completed his music and program, reporting of how proud she was, and excited to see Yuri's program too. She's probably trying to fish out details from his program. He does up the braids he would be doing for his Free Skate and sends that picture to her, a trade that he figures doesn't let out anything substantial on his side. Yuuko just replies a flurry of praises on his beauty. Despite himself, Yuri smiles at this easily captivated lady, and her enthusiasm to just be excited over every detail about him in a caring way. It's quite unlike that rabid fan group he has. They are just terrifying.
Yuuko texts him again after assignments were released, after Yuuri wins regionals, before Skate Canada and after Yuri himself wins silver there, one time when the triplets found a picture of him photoshopped into all Hello Kitty print, a few nights before Rostelecom, and some more in between. Every time, she is all care and enthusiasm, and honestly, it's both overbearing and infectious. He wants to dismiss her as some annoying spy for Yuuri, but it's just been very uplifting instead. And uplifting is so foreign a sensation for Yuri. He's had satisfaction, fulfilment, elation, but that's when he wins. Uplifting is new, and he kinda likes it.
Yuri still isn't used to receiving care, though. Grandpa isn't here, and now not only does Yuuri win a personal best and come so close to a new record, he and Viktor both cheer for him? What a diss, Yuri doesn't need your care. He stumbles on the first triple Axel, but picks himself up again and delivers a good score. When Viktor has to leave for Japan, and Yuuri ends up affected, Yuri knows this is not how he wants to win, not off the failure of a man who was capable enough to win Viktor over. He actually cheers for this emotional wreck. Though JJ catches him and it's enough to put him off showing any semblance of concern for the rest of the competition.
Yuri has to run away from the walking emotional disaster zone after the awards ceremony, he is not a hugger. But he also sees something in Yuuri that evokes a heaviness that settles in his heart, though he doesn't know what it is, this feeling. He's feeling pretty salty about losing the gold to JJ twice now, but Yuuri moping about like this also occupies him. Maybe it's the disappointment of not doing as well as he could have, Yuri would feel pretty angry at himself to messing up, and he's seen enough to realize that Yuuri tends to go with sadness instead.
So Yuri, picks up the bag of katsudon piroshki his grandpa made him, and sets out to find Yuuri. At least he'll be able to eat some katsudon with Viktor not around, and maybe that'll put him back on the mood to skate well. Yuri does not want to win against Yuuri like this again. It makes his heart heavy, and the feeling itself unsettles him too.
When Yuuko texts him congratulations again the next day, Yuri decides to ask her about his that feeling with Yuuri. He figures that since she's got him into a new state of feeling, she'd know about this one too. "So silly," she replies. "You feel that way because you care about Yuuri. I'm so glad you do too! What you did for him was very nice. You're a great friend to us all! Good luck for the Grand Prix Final!"
Yuri is in slight disbelief to learn that he cares for Yuuri. He buries the thought. That's absurd. He only doesn't want to win against a poor performer, because it makes his win look cheap. That's gotta be it. Yuuri's got more than enough care going for him, he doesn't need any more from Yuri, and he's not getting any. Nope, Yuri Plisetsky does not care for Yuuri Katsuki.