if you didn’t commit to spending the entire evening with her on her birthday
Bad news. This is sound logic, but in NT world, there are all sorts of implied social contracts. Dating someone defaults to "yes you DID commit to spending the evening with her on her birthday, unless otherwise discussed to opt out".
Not everything needs to be spoken or written down. If I walk into a restaurant but there is no sign that says "please form a line to place your order," I'm not going to barge to the front and begin ordering, because "form a line and wait your turn" is understood to be how society functions.
You are absolutely not obligated to follow the implied social contracts. And you not obligated to know all of the implied social contracts. But you also don't get to take the moral high ground on the situation.
she doesn’t have a right to blame you for her hurt
A healthy relationship doesn't have blame or a scoreboard or anything like that. It REALLY doesn't matter who is to blame. Try to win an argument by saying "Well you dont have the right to blame me." It will end more relationships than it will win arguments. If you value the relationship, you want your partner to not hurt regardless of the source/blame. The hurt happened and all you can do is figure out how to prevent the hurt going forward, which will often be by communicating and setting expectations.