Norio said this argument was Anna's fault. The only fault I find on Anna's part is that she's a teenager that hasn't matured yet and acts kinda spoiled, on top of being generally energetic and nice to basically everybody, singing their praises without giving it a second thought and to the point of being annoying sometimes.
For someone whose life is about repressing their wishes in favor of their loved ones, family or otherwise, having a spoiled girl compliment you at every turn might sound condescending, but this is where I'm going to place the blame on Moeko. She clearly said to Anna that she should've spoken right away if she was upset about something, but what about her? She said Anna complimenting her feels annoying, which implies this is been going for a while, why she hasn't said anything before? Why build up resentment only to unleash it all at once and hurting Anna in the process instead of helping her understand where the problem lies in? Supposedly Anna is her friend, right? She should know better that Anna is not the kind of people that looks down on others, but still she has the nerve to accuse her of being fake? Of course Anna is going to hesitate to repeat the compliment right after her "friend" accuses her of not being sincere.
I don't know if Moeko is envious of Anna and thinks that she has been had everything in a silver plate but just talking about getting a boyfriend, Moeko has no right to be jealous of Anna. Anna has been proactive in her pursuit of a boyfriend, even if the process to get there seems like it just happened, and actually tried to make some moves, and what's Moeko done? That's right, interfering with Anna's moves but more importantly she has used Anna, that's right, she has used Anna to bait for guys and tried using Anna to get someone's Line ID. Honestly Moeko had shown how she really feels about Anna, all good when it's convenient for her, annoying when it doesn't suit her, that honestly isn't a friendship worth keeping.
I hope Anna grows from here on and stops being clingy to everybody and just be clingy to Kyotaro, also to keep some distance from Moeko, a person that tries to help you grow and fix yourself is a good friend. A person that sees some faults in you yet instead of helping you overcome those faults uses those faults to trip you and call you a pest isn't a good friend.