What criteria have to be met to term a carboxylic acid a 'fatty acid'?
I suggest to consult the IUPAC Compendium of Chemical Terminology, aka the Gold Book for a definition of fatty acids!

In a narrow sense, fatty acids are defined as


  aliphatic monocarboxylic acids derived from or contained in esterified form in an animal or vegetable fat, oil or wax. 


The definition continues to mention that  


  natural fatty acids commonly have a chain of 4 to 28 carbons [&hellip;]


The paragraph closes with a wider definition, namely: 


  By extension, the term is sometimes used to embrace all acyclic aliphatic carboxylic acids.




Personally, I have used the term fatty acids only for natural fatty acids.