These are my thoughts following up on the ending of episode 11, and the beginning of episode 12, which I believe, further explain why this scene emphasises and reminds us of Yuuri's kind and caring personality, and explains how Yuuri searches for a mutual agreement between himself and Victor. Brought about by @talesofcolor, who wanted an expansion upon the previous analysis.
Something important to remember is that being selfish does not only mean doing what's best for yourself without considering others, it can also be thinking you know what is best for another person, and then acting upon your own assumptions and thoughts without consulting the other party. I think this is what Victor means when he specifically says that Yuuri is being selfish about this decision.
Yuuri's version of him being "selfish" is very strongly tied to his personality, and more specifically, his anxiety and low self-esteem. While in reality, Yuuri brings up this idea for Victor's sake, so that Victor may continue his career, in his mind, he may have twisted it so that he thought that it was in fact himself who wanted to see Victor skating competetively again. It's his anxiety working against his thoughts, and his low sense of self that makes Yuuri believe he is being selfish in this situation, hence why he too says that he's being selfish. Also being the more sensitive and emotional person in this pair, Yuuri may be more easily emotionally swayed, and can empathise with others easily when in these types of highly emotional confrontations.
When you get down to it, essentially, Yuuri has decided to come to this conclusion, and discuss the options with Victor because he cares about him. Yuuri knows that by keeping Victor as his coach, he is "killing" his skating career, even Yakov commented on this at the very beginning of the anime. So, Yuuri comes to the conclusion that the only way Victor can continue his competitive skating career is to have him resign as his coach (for now at least), and he brings up this discussion with Victor, providing his own reasons, while also listening to Victor's own reasoning.
I said earlier that Yuuri wants himself and Victor to come to a conclusion to this situation together, and that he doesn't want to, or at least isn't intentionally, forcing the idea of resigning upon Victor, and this can be seen in how Yuuri still doesn't push for an agreement right there on the spot, he allows them both to further think about the idea of resigning and their future, and they decide to come to their own conclusions about their own respective futures.
ăăăȘăŒăç”ăăŁăăăăăăăèȘćă§çăăćșăăšæ±șăăă
(furi- ga owattara, sorezore jibun de kotae wo dasu to kimeta)
"It was decided that after the Free, we would give our own respective answers"
Victor decides whether he resigns as Yuuri's coach or not, and whether he makes a comeback or not.
Yuuri decides on whether he retires as a competetive figure skater or not.
Neither will make a decision for the other, that's what this line infers.
Another nice thing to note is the section ăçăăćșăă(kotae wo dasu) which can mean "to state one's answer" and can also mean "to work out a solution". While the latter does this better than the former, both of them infer that even though they will each decide for themselves what they will do with their future, they will tell each other what they will do next. It is a prompt for communication, and that's what happens later. When each of them come to their own conclusion on what they will do next, they immediately tell the other. To me, it is just refreshing to see a couple represented in anime that is honest to both themselves and each other.