I would agree with "award document" being the simplest and most descriptive term in one.
But I don't agree with you on the second point and I think the British argument is quibbling - because their medals weren't "official" until gazetted - an official announcement - i.e. a piece of paper. I would argue that the London Gazette announcement or any piece of paper issued by the Crown was the "award" not the medal inscribed with a name.
And using your Kaiser logic above, the Kaiser didn't say, any actual EK given out is "official," he said that any document given out by an authority is the Besitzzeugnis. Ergo, only the document is official; not the medal itself. The Besitzzeugnis is what makes the medal "official." Therefore, the document, as official proof, is the award.
This extends to the American system, in which case neither the medal itself nor the certificate is the official award. You can get both in a nice ceremony. But the "award" is the actual orders announcing and authorizing the medal - the only official document in the group (even the certificate will usually refer to the orders number). At least this is the modern system; I'm not sure about WWI, but I'm willing to bet that the General Orders announcing the award was the true official document and therefore, the "award" itself, as medals probably could be obtained from sources other than the US Government.