Why so ashamed? - she pushed you and pushed you and learned that she can only push so far. Good lesson for a child particularly as an apology fixed it.
Personally I would have put her in her room, close the door and sat outside holding it shut until her tantrum broke - you can then open the door and say "when you calm down" "when you do what Mummy has asked". I have never held with 'she refused to stay in her room'
She was being a challenging brat! You are being a parent.
Kids need parents who will lose their tempers and be normal with them IMO