I think I understand the distinction but for the different versions of the Tourists deck, you say that the Hochman US9 is scarce only because of the number of prints or also due to its age and because it's the first version please ?
Quantities printed on older decks aren't always available - I'd say the public discovers them infrequently to rarely, in fact. Scarcity it just that, they are few known to exist. A few thousand or a few hundred thousand might have been printed, but today, for a variety of reasons, few are left. There are almost countless scenarios explaining why a vintage or antique deck would be rare - or common.
For the Circus deck, it's rare because the deck was aimed at a children's audience, perhaps even sold at actual circuses. Just as few of their comic books, board games and toys survive, so, too, do few of their decks of playing cards.
For something like the War Series, they didn't get printed until the year the war ended, so someone at USPC made an executive decision and ceased production on the decks, giving them an unusually short print run. The decks were being promoted to be played with, so I'm thinking some decent amount of that production run was lost to wear and tear. Thus, another rare deck (four decks, actually, not counting color variants).
Certain souvenir decks aimed at adults (like the Brown Derby deck) would have two factors determining surviving quantity - they'd be scarce because there were few places where they could have been purchased in the first place, thus probably had limited distribution, but the decks that were distributed would have had a better chance of survival because souvenirs of that type are often left unused, either displayed on a shelf or dropped into a drawer and buried in the back of it.
Then there's the Bicycle Rider Back - undoubtedly the most common Bicycle back printed, probably in continuous production since it was first introduced in 1893. Sure, many got used up, but so many more were made that they're all over the place. Plenty of old decks still exist.
You can, see, though, there are factors that come into play besides the size of the print run, a number that's often lost to antiquity.
