Personally may not be the best word, since as you point out, they aren't meant about employees in general. I think that when they say personally, what they mean is either she gets defensive or offended to the point where the opportunity for a discussion about how to improve is lost.
For example, if someone says that a report is sloppy, taking it to mean you're sloppy and responding along the lines of no, I'm not sloppy, I am always careful or no, it's great, I don't know how you can say that would be examples of what I would call taking it personally. A better approach would be to say, I'm sorry that it wasn't what you wanted, can you give me some examples of things that need revision and how you'd like them revised. That opens up a conversation along the lines of this wasn't spell checked, I think some of the wording is awkward, etc.
Another example might be if someone says you take feedback badly. If the response is no I don't or I can't believe you'd think that one might refer to it as taking it personally even though it's not the best word. A better response would be, I'm sorry I gave that impression, I do value your feedback. Can you give me examples so that I will have the opportunity to improve. If you honestly believe the comment isn't justified, you could add, I do value your feedback and do my best to take action on it, for example, when you told me XYZ, I did ABC to improve.
In this case, if she absolutely felt that she had to respond, she could have said, "I couldn't help but overhearing the end of your coversation and am sorry to hear that you think I take things too personally. Can we set up some time to talk through some examples and ways I might improve". Although she would have to admit to overhearing the conversation, at least it states a willingness to change.