How is 'everyone an authority' different from 'there are no authorities?'

Well, using your definition of authority as "person with opinions that others agree with", these statements would translate as follows:
- 'everyone is an authority' becomes "Every person has opinions that others agree with."
- 'there are no authorities' becomes "No person has opinions that others agree with."

The problem is that you seem to want to use the connotations of the word "authority", but you aren't explicitly including them in your definition.

What's your definition?.

I don't use the word 'authority' in reference to people, because it communicates meaning badly. I'd prefer to use a word like 'expert' or a phrase like 'informed on the subject'.