I think Bertrand's point, about the "a" meaning "with," is not minor but quite significant. If Giusto commissioned a deck of naibi with triumphs, he commissioned a deck made up of the common regular cards plus the perhaps uncommon triumphs. From Giusto's use of the word "naibi" for the regular cards, which were used in the inns, it then doesn't follow, nor is it even suggested, that the deck that is a combination of the two, were used in the inns. The combination of common naibi and (i.e. with) possibly uncommon triumphs may well have been uncommon.
And in fact, it would be strange for Giusto to be giving Malatesta merely an expensive version of something common, as I think Girolamo suggested. That would be like giving Malatesta a set of ivory dominoes with his coat of arms on them, or a pair of gold-plated dice: something which Malatesta would rightfully take offense at, as though saying, "you're rich and powerful, and deserve rich gifts, but you're really just an uncultured commoner."
No, it is more likely that the combination common cards plus triumphs was fairly rare, a new fashion that had recently taken hold, probably, and one that connotes high society. That might have meant a game up to then based in the nobility, and thereby a status-symbol among those who weren't "old blood" like the Visconti and the Estensi. It is possible that Malatesta would have considered the Signori of Florence high enough society; I don't know. But I'd guess that something the Visconti had would have been even more desirable (and even more if Visconti's condottiere had it, too,).
To be sure, Florence was likely in the forefront of producing triumph decks in the 1440s and certainly the 1450s, a prominence that may have been taken over by Bologna later. But the fact of production does not mean origination. Florence was a town of wealthy bankers and merchants who were also art patrons. There was a base of skilled painters there that made it at a natural base for card production for the affluent but untitled. Someone could bring a design in from elsewhere, even borrow a deck from someone, and have a deck done with those concepts.
Before Sept. 1440, we have no idea. The new fashion may have come out of the Conclave of 1438-1439, where people might have been exposed, from members of the Ferrara court or from Lombard or Ferrarese-based attendees of the Conclave, to a game that up to then had been mainly an educational one in the principles of Christian life for women and children of the nobility and little regarded otherwise, and one which boys were expected to forget about in favor of chess by adolescence.. But as such it might have been a game supported by the clergy in the lands from Milan to Ferrara, And perhaps some recent rule changes made the game more complex and of more interest to educated adults.
For Ferrara before 1440, we have the evidence (http://trionfi.com/playing-cards-ferrara-1422
) of the 1422 document of 13 "newly made" cards along with the repainting of four "para", which I take probably to mean combinations of number and figure cards, i.e. suits ("para" meaning here originally, "combination of two"; see Florio's entry at http://www.pbm.com/~lindahl/florio/search/370r.html
), The 13 cards are probably not tarots, because they are described as 5 "figures" plus 8 other cards. If they were tarots, they would presumably all be called figures. (However I am not sure about what exactly "figures" means.). In any case, they are likely a fifth suit, which is what triumphs became, and the painter "Iacomo" is likely the Iacomo Sagramoro who painted the "triumphs" of Feb 1442 and the "14 cards" of Jan. 1441. So this is some weak evidence of tarot or proto-tarot activity in Ferrara.
For Milan, there is the evidence of the Michelino, a proto-tarot in that it had what appeared to be 16 special cards that would have trumped the regular cards. There are also the devices on the canopy of the Cary-Yale Love card, which look like they might have originated in a somewhat similar card in a deck done for the 1428 marriage of Filippo Visconti and his new Savoy duchess. (The Cary-Yale Love card itself of course has other details that suggest a different marriage.) Moreover, the clothing of the Cary-Yale and Brera-Brambilla are similar to that in sketches done c. 1430 by Pisanello. None of these suggestions is at all conclusive, to be sure; all could have other explanations. But they are suggestive.
Bologna has the 17th century painting with the sentence written underneath, containing several verifiably false statements, but declaring that the "tarochino" was invented by Francesco Fibbia, who died in 1410. This Fibbia actually existed, as Vitali's research indicates. This writing is to be sure also is merely suggestive.
Concerning Florence, there is the Ferrarese document that the Parasina d'Este arranged to have "brought from Florence" a pack ("paro") of "VIII Imperadori" (http://trionfi.com/imperatori-cards-ferrara-1423
), a possible ancestor of the tarot. It also suggests that Florence was even then a producer of hand-painted specialty cards. Likewise, the fact that Florence produced a deck with triumphs in Sept. 1440 does give some credit to the suggestion that they originated there, since it is the earliest dated document known. However the circumstances of the gift also point in the other direction, that the deck of common cards plus triumphs originated in the courts of the Emperor-sanctioned hereditary nobility.