I'm not sure what in the name "fanfilm" would suggest its status as such would be dependent on the number of people involved in making it. It's a pretty self explanatory descriptor
Actually, I'd argue it's not. The descriptors are getting murky.
Is "Renegades" a fan film? Is "Of Gods and Men?"
Is Phase II still a fan production? Is Star Trek Continues? Is Starship Farragut?
At what point does having professional involvement push a fan film out of that arena and into a semi-professional or even professional production?
A fan film will always be a fan film unless it is produced and distributed via official channels. The fact that professional actors and directors are attached to it doesn't mean anything to the label "fan film". Even if they threw 150 million at the wall, it would still be a "fan film".
The labels here have to be distinguished:
fan film - official film - independent/original film
no budget - low budget - high budget
Of Gods and Men is a fan film. Star Trek Into Darkness is an official film. Polaris is neither, it's independent or original.
The budget question is totally independent from "fan" or "official" films. Official films can be no budget, and fan films can be high budget.
"professional production" is a strange term. When is a production professional? Give James Cameron a camcorder. Will he be doing something professional with it? Give a bunch of students the most expensive equipment. Will that make their attempt professional? If professional actors, writers and directors come together to make a film in their garage as a hobby, is it a professional production?