I think you’re both in agreement. “My code” when it’s a bug is fine because you’re really showing humility and responsibility. I think it is the one exception to “our code” being used as the standard way to refer to code. When the code is great it should be “our code” and by proxy the team’s success. Not one person’s. And when there’s a bug that’s not your own it should be “our bug”. You don’t want to throw anyone under the bus because we all make mistakes. (That’s not to say all bugs should be ignored. Your team should investigate why their code failed and how they collectively can learn.)

I think the “my code; my bug” thing is great because if you lead by example people will follow suit and take responsibility for their failures as well. I think that’s a great effort to move towards a more positive culture.