"Official" is produced and sold by a postal agency and that should be indicated on the cover. "Un-official" is manufactured and sold by a private company - and again - their name and / or copyright should appear somewhere on the cover.
An unofficial FDC may also be a home-made cover with any marking to designate it as day of issue. A cover post-marked on the day of issue (with nothing to indicate it) is also a first day cover.
I look at it this way at least for the present. The US Postal service issues uncacheted stamps with first day of issue. Those are the ones I feel are official. The others for me are the one I produce which could be someone else (Gamm) which I create a double cancellation using as previous cancelled cover such as the the Sleeply Hollow and the new Halloween Stamp.
Your question is too vague to give an absolute answer.
If we are talking about the "cover" more so than the cancellation: An "Official" cover would be a cover produced by any organization that puts their name to it, or allows their name to be mentioned on the item - it is not a monopoly by postal agencies. It is just a title and nothing more. It may be limited to a single or many products.
Uncacheted FDCs are in no way "official" - they are blank. The USPS, or you, can make them.
Then comes the cancellation aspect which XNBer alluded to. An "official" FDOI cancellation is any device produced by a postal agency explicitly to be used on the day of the first day of sale and at the location(s) designated by the postal agency as "official" locations. For the U.S., you would have the machine FDOI cancel, the hand FDOI cancel, the Bullseye hand canel (without slogan), the black/white pictorial cancel, and the Digital Color Postmark. Any other cancellation in regular use at a postal agency would be considered an unofficial First Day location. As XNBer stated, if the postmark is on the First Day, it is a FDC no matter the location of the postmark.
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