> At what point does a prop become a fan?

Linguistically, a prop becomes a fan when you anchor it to something immovable, or attach it to something movable but without any intention of propelling it using the prop.

Fan - try to move the air around, but not the thing it's attached to.

Prop - try to propel the thing it's attached to

Edit: I realize that might sound ambiguous given terms like "turboprop" and "turbofan" both of which are on planes.

The turboprop uses air to speed up propellers to move the plane. A turbofan uses just accelerated exhaust from the engine, and the fans are just a means of feeding the engine combustion material (air)