I assume that you are looking at used cameras, in which case why does it matter?
Any warranty will be with the seller and there is nothing else that will affect your use of the camera or its maintenance, servicing or repair.
Speaking with Olympus, yesterday, I was surprised to be told that any remaining warranty transfers with the lens. So if it was registered with three years' warranty, then an 'authorized' second-hand lens could have residual manufacturer's warranty, whereas a 'grey imported' lens would only have whatever warranty the retailer offers, plus your statutory rights under the Consumer Rights Act, 2015.
When I asked Olympus what identifying marks would be on an Olympus 'authorized' lens or its packaging, they had no answer. They asked me for the original Bill of Sale, so they could verify whether the retailer is on their list of authorized sellers.
So, it is appropriate to ask this question, as Olympus do not (currently) offer a global warranty service.
From my point of view, I buy grey imports if the price is right. A £1,134 'authorized' camera body vs £430 for the same item as a 'grey import'. Gives me a lot of cash to cover any future repair costs, buy insurance, or simply buy another camera if anything goes wrong.
In any case, many grey importers also have a reputation to maintain. You'll often get a 2, or even 3, year warranty from the importer. Not to mention, if you buy by credit card in the UK, you're covered by the Consumer Rights Act, 2015.
Olympus also tell me, you can pay an up front surcharge for them to repair your grey import, and the figure quoted was far less than the price difference between grey and authorized cameras or lenses.
If "The Customer is King", then Olympus need to get their act together, join the rest of us in the Global Marketplace, and start offering global warranties, as others do. After all, grey import sales must constitute the vast majority of Olympus Worldwide corporate sales revenue.
Rascal