Should the "about" section on a LinkedIn page be written in the first or third person?

I agree that it should be in the first person. It is an overview of your professional accomplishments written by you. There is no illusion that it is about you as written by someone else (as a professional biography would be on a company website or professional journal).

Third person would read strangely to cookieman and cookieman would worry about that person.

Mine is neither. Both options waste words, in my opinion. Mine says:

A doctoral candidate, research associate and undergraduate instructor in [this department] at [this university] interesting in [these things].

I had to write mine based on a professional development-based person in my field and online modules in a class I took as part of my PhD training. Not saying it’s right, but this is what we were trained to do. No need for complete sentences, names, or pronouns… just get to the meat of who you are and what you do.