I read it the way its supposed to be read.
That sentence doesnt say that atopic eczema is caused by a deficiency in fatty acids. Its says its "POSSIBLE that the dry skin of atopic eczema is deficient in certain fatty acids". Not that deficiency in fatty acids causes atopic eczema.
I havent read any more into the subject but based on the statement it could just mean that dry skin deficient of certain fatty acids are a symptom of eczema. Like dry crusty skin is a symptom of EC. That dry crusty skin isnt the cause of the EC. Its the inflammation that caused the dry skin which is deficient in essential fatty acids.
But for the record I do believe that the two are relevant to each other because they are both inflammatory skin diseases.
And Longterm I dont think you have EC. Just because your symptoms are SOMEWHAT similar (not even that similar) doesnt mean you have EC. Things that seem to work/improve the lips of people with a condition like yours, seems to be very different from things that seem to work/improve the lips of people with EC.
I think that they are two different forms of cheilitis that are probably treated differently.