Managing relationships with uninvolved supervisors
The main problem from your side that I see here, is that you published the article without even telling her in advance. (Which might be based on you seeing her so rarely.)

Had you asked in advance, and maybe told her that this is how things work at your home university, she would probably have given her consent. That way she might feel backstabbed.

So the immediate action for you should be an appology from your side, telling her, like you told us, that this is what you thought you were supposed to do. (But don't make it a pure explanation, but make it an honest appology, the phrase "I am sorry" should appear at least once.)

I never had a publication without my advisor during my PhD, and (since we did alphabetical order) he was first author of all the papers. But than again, I met him on a weekly basis, so he was far more involved. I assume that by now, things have gone wrong for too long to change without you havily interfering (by requesting more frequent meetings and more involvment from her side). However, if you want this, I would tell her in a seperate discussion.

Thus, from this short description, I see the main flaws in your relationship on her side, but the main need for a concrete appology on yours.