There are some grey areas, here. After this year's Worldcon is over, I'll pester next year's Hugo committee about these questions.
What is a fan artist?
As far as I can tell, a fan artist is a science fiction or fantasy fan who produces art. As far as I can tell, fan art can be in any medium and can be original or derivative.
As far as I can tell, a fan artist does not have to be an artist who produces art related to copyrighted work or franchises like Dr Who, Harry Potter, a Game of Thrones or Lord of the Rings, though those artists are certainly included. I think that the copyright holders usually object to derivative artwork if it is for sale. I think that they generally don't care if the artwork is not profitable.
What is a science fiction or fantasy fan?
Interesting. Fans are defined for such awards as the Australian Ditmars as "natural persons being active in fandom." Which usually means they have attended or otherwise been involved in fan conventions, production of fanzines, or genre-related websites. Hence the inclusion of artists who exhibit at science fiction or fantasy conventions.
What is a semi-prozine? (ie. what constitutes poorly paid?)
Fan art must be unpaid or poorly paid, such as in fanzines or semi-prozines. But where is the line drawn between semi-prozines and prozines when it comes to cover art or internal illustrations? SFWA defines a prozine by the amount per word paid to its writer contributors, but I can't find an amount to be paid to the illustrators.
Clarkesworld is considered a prozine by SFWA, a semi-prozine by past Hugo committees, and it pays $100 for a cover illustration. Apex magazine, also somewhere on the border of pro and semi-pro, pays $50 for digital reprint rights for cover art.
Professional artists might ask for something in the order of $1000 and up for an original cover that would make them eligible for the Professional Artist Hugo Category. But it's rude of me to start demanding to know what people have been paid, and so I'm not sure what to do about this rule except to err on the side of caution.
And what about those aforementioned fan artists exhibiting at conventions? What happens when their work sells at the end-of-con auction for more than they might have received from a semi-prozine? How much money to they have to make, or what percentage of their income must be from art, before they are shunted up into the Pro Artist category?
The Pro Artist category definition is "for artists of works related to science fiction or fantasy released in the previous calendar year."
Which includes all the fan artists, doesn't it?
ARGGHHH BRAIN HURTS.
Please correct me in the comments if I've said or assumed wrong things. Like I said, I'll start harrassing actual knowledgeable people once Chicon 7 is over.