I'd have thought from Kodak's position as the licensee it makes sense to claim that Endura is theirs Isn't the analogy the same as that of Ilford chemicals Someone else makes them but Ilford holds the licence and puts its name on the product?
It works the other way around. If someone licenses the Kodak brand name, it's that 3rd party (the licensee) who decide what product they put it on (within the terms of the licensing agreement) and how that product performs. So if Kodak licensed the name (in this case, 'Kodak Professional') to a licensee, it's that licensee who are responsible for the product as such.
This is different from outsourcing manufacturing and then selling under one's own brand, as Ilford apparently does with their chemistry (assuming they do; I never looked into it). The example of Ilford color film was really more appropriate here than that of Ilford B&W chemistry: a product marketed by a 3rd party with a brand name pasted onto it based on a license agreement with the company holding the rights to that name.
So is it KA who have commissioned the resumption of this paper knows as Kodak Endura thus we should refer to it as Kodak Alaris Endura if Eastman Kodak of Rochester has no involvement at all with this paper?
That's unclear at this point. In the case of Endura paper, it has been SinoPromise over the past 7 years who decided what product would be sold as Kodak Endura. Not Kodak. Maybe this has changed; see Matt's post above. SinoPromise has never been very outspoken on what's happening on their front, especially not in the past couple of years.
AFAIK this particular product is known as 'Kodak Professional Endura Premier'. At least that's what's on the label. We didn't call it 'Carestream Kodak Endura' or 'SinoPromise Kodak Endura' in the period of 2016-2022 either. The product name as such seems to not have changed. The product has likely undergone many changes over the year, despite being called the same. Even if it has always been manufactured by the same entity. Incremental, but subtle changes to color paper are rather common.
What is clear if that someone has signed off on the costs of restarting production, assuming of course that production has in fact restarted and that we are being told the whole truth
Well, apparently back in September 2022 someone had signed off on production costs. I'm not entirely sure what the relevance is of knowing if production costs were signed off on, and by whom. It seems to me this is mostly relevant to the controllers and accountants of the company or companies involved. If product leaves the factory, as an end user I always assume that someone has paid for the materials. If they haven't, they managed to get a very sweet deal indeed. I'd still be none the wiser from my perspective.