It sounds like a perfect example of her communicating something to you, but you are not hearing it. Or, you are not hearing it correctly. I don't think its cultural. She is telling you that she doesn't want to be obligated to call you every day or every week. She is telling you that your desire for more frequent communication is a source of stress for her. I think you should believe her, and give her some more space. And, it sounds like you already agreed to give her some space and for you to become more independent. This is good for both of you. Now it is time to become more independent and stop looking for ways to interact with her and change her behavior.
And ask yourself; would I still feel connected to her even if we only chatted once a month, or once every two months? If your answer is yes, then great. If you must have more frequent communication, however, then this isn't the girl for you. There is no magic phrase in English or Japanese that will suddenly make her communicate more frequently with you.
Let me say another way: you are trying to force her to do something she doesn't want to do. It is not healthy.
You know the cliche: if you love someone, set them free. One way to show her how much you love her is by giving her the freedom to call/contact you whenever she likes, even if its only once a month.
Regarding social media frequency and cross-posting and tagging and liking. Ignore it. Social media is not real life. I know it is too tempting to analyze her social media life, but really it is just pixels on a screen, and you shouldn't obsess over it. If she thinks you want to control her "reposting" she will be rightly annoyed. We can't help but become jealous sometimes...but we can understand jealousy as a toxic presence, and try to moderate its influence.