The following site has transcriptions of the known letters of William Adams (the inspiration for Blackthorne):
From the earliest letter dated October 1611 (highlighting mine):
Being at the island of St. Maria, which lieth in the lattetude to the s°ward of the line of thirtie seuen degrees twelue minutes on the cost of Chili, wee tooke counsell to take all things out of one ship, and to burne the other; but that the captens that were made newe, the one nor the other, would not, so that we could not agree to leave the one or the other; and having much cloth in our ships, it was agreed that wee should leaue the coast of Perow, and direct our course for Iapon, having understood that cloth was good marchandiz there ; and also how vpon that coast of Perow, the king’s ships were out seeking vs, hauing knowledge of our being there, vnderstanding that wee were weake of men, which was certaine; for one of our fleet, for hunger, was forced to seeke reliefe at the enemies hand in Saint Ago. For which reason, hauing refreshed ourselues in this Iland of St. Maria, more by policie then by force, we departed the twentie seuen of Nouember, from the Iland of St. Maria, with our two ships ; and for the rest of our fleete we had no newes of them. So we stood away directly for Iapan, and passed the equinoctiall line together, vntill we came in twentie-eight degrees to the northward of the line: in which lattetude we were about the twentie third of February 1600. Wee had a wonderous storme of wind, as euer I was in, with much raine, in which storme wee lost our consort, whereof we were very sorry: nevertheless, with hope that in Iapon we should meet the one the other, we proceeded on our former intention for Iapon, and in the height of thirtie degrees, sought the northermost [?] Cape of the forenamed Hand ; but found it not, by reason that it lieth faulce in all cardes, and maps, and globes ; for the Cape lieth in thirtie-fiue degrees 1/2 which is a great difference. In the end, in thirtie-two degrees 1/2, wee cam in sight of the lande, being the nineteenth day of April. So that betweene the Cape of St. Maria and Iapon, we were foure moneths and twentie-two daies; at which time there were no more then sixe besides my selfe that could stand vpon his feet. So we in safetie let fall our anchor about a league from a place called Bungo. At which time cam to vs many boats, and we suffred them to come abord, being not able to resist them, which people did vs no harme ; neither of vs vnderstanding the one the other. Within a 2 or 3 daies after our arivall, ther cam a Iesuit from a place called Langasacke, to which place the Carake of Amakau is yeerely wont to come, which with other Iaponers that were Christians, were our interpreters, which was not to our good, our mortal ennemies being our Truchmen. Neuerthelesse, the King of Bungo, the place where we arriued, shewed vs great friendship. For he gaue vs an house a lande, where we landed our sicke men, and had all refreshing that was needfull. We had when we cam to anker in Bungo, sicke and whole, foure and twentie men, of which number the next day three dyed. The rest for the most part recouered, sauing three, which lay a long time sicke, and in the end also died. In the which time of our being here, the Emperour hearing of vs, sent presently fiue gallies, or friggates, to vs, to bring mee to the Court, where his Highnes was, which was distant from Bungo about an eightie English leagues. Soe that as soon as I came before him, he demanded of me, of what countrey we were ; so I answered him in all points ; for there was nothing that he demanded not, both conserning warre and peace betweene countrey and countrey: so that the particulars here to wryte would be too tedious. And for that time I was commanded to prisson, being well vsed, with one of our mariners that cam with me to serue me. A two dayes after, the Emperour called me agein, demaunding the reason of our comming so farre. I aunswered : We were a people that sought all friendship with all nations, and to haue trade in all countries, bringing such merchandiz as our countrey did afford into strange landes, in the way of traffick. He demaunded also as conserning the warres betweene the Spaniard or Portingall and our countrey, and the reasons ; the which I gaue him to vnderstand of all things, which he was glad to heare, as it seemed to me. In the end, I was commaunded to prisson agein, but my lodging was bettered in an other place.
(Note to mods: this is taken from a book published in 1850, so there should be no copyright issues involved.)
It’s clear that by 1611 Adams is using “Japan” (allowing for spelling variations) rather than “the Japans”.
Going back further, the Theatrum Orbis Terrarum, a world atlas published in 1570 in Antwerp (so clearly well know to the Dutch) labels Japan as “Iapan”:
Conclusion: “Japan” was well known usage at the time Shogun is set, at least amongst the literate. Possibly “the Japans” could have been “lower deck” (uneducated) usage in a semi-mythological “cities of gold” sense to encourage the crew, but there is no way to confirm that.