I'm sure I would have felt similarly, initially -- but I think the difference is that if my friends/family reassured me that it wasn't a big deal, then I would trust them on that and wouldn't beat myself up about it too much. (But if they seemed like they didn't have a good time, I would have reacted exactly the same way that you did!) I'm guessing this is so-last vs. so-first, i.e. if external validation wouldn't have mattered to you, in deciding the severity/importance of what had happened with the cake.
Edit: That was presuming that your family was understanding and nice about it -- which I probably should not have presumed. I hope they were understanding and nice about it! Were they?