I suppose we all have our own definitions. Here are mine:
A fan-edit is when someone re-edits a film in an effort to improve on it by making changes, or combining films in various ways to make new films that didn't previously exist. Adywan's SW Revisited and the Phantom Edit are examples of fan edits.
A preservation is when someone takes an existing film and digitizes it so that it can be made available for future generations, sometimes accompanied by cleaning or other fixes to counteract the deficiencies of the previous media. DJ's trilogy and the Puggo Grande are examples of preservations.
A restoration is when someone rebuilds something that once existed but no longer exists or is no longer available, out of whatever can be found that is similar, or by rebuliding missing components. Harmy's is an example of a restoration, because the theatrical version doesn't exist in HD and he has tried to rebuild it out of other parts.
My definitions of "preservation" and "restoration" come from how those terms would be used in the antique business. Sometimes restoring a piece of furniture requires replacing pieces of it with new wood. Thus, I don't consider Harmy's to be a fan-edit. His "partly despecialized editions" that preceded his current work were fan edits, but his DEs are restorations, in my opinion.
It is hard to categorize Lucas' special editions. They are sorta like fan-edits, but not made by fans. They aren't even directors' cuts because he wasn't the director. Maybe "executive producer's cuts" ??