Disclaimer: My knowledge of Chinese/Japanese and Korean early printing practices is extremely rudimentary, so I can only offer limited detail on these types of texts - in my line of research, I work with early printed books in England, and with early modern manuscripts. I can speak much more effectively to the Gutenberg side of this question than the Chinese/Korean printing part. The following summary of Asian printing comes from my work in teaching undergrad History of the Book, so it's not terribly detailed, but hopefully offers a decent enough overview of the issues from a reliable source, and with a bit of luck, other, more qualified folks can fill the picture out a bit.

To begin - the assertion that the Gutenberg Bible is the first printed book is, as you have already identified, problematic. It was the first book to be commercially produced in this way in the West, but by no means the first book to be printed with movable type. Claims to the contrary are seated in a deeply Eurocentric view of history. As for how this reputation comes about has a lot to do with the characteristics of Western writing, and the kind of religious and commercial environment in which Gutenberg worked.

So - early Chinese texts. By the seventh century, woodblock printing was in use in China, with movable type by the eleventh century. One of the most remarkable early printed texts is the Diamond Sutra of 868, regarded as the earliest complete printed book. The British Library holds the document, and you can see more about it here. It's presented as a scroll, not a codex (that is, what you or I would immediately identify as a 'book') but has plenty of features that would become standard here in Europe centuries later - woodcut printed images are used, alongside printed text, and a colophon describes the circumstances and date of printing. Early Chinese printing was xylographic - that is, employing carved woodblocks to 'stamp' the image rather than cast metal type. While wooden blocks are not necessarily as durable as metal counterparts, they offer some advantages, one of which was the ability to produce blocks in the author's actual handwriting, which was popular for the prefaces to printed works. One of the big impediments to printing in a non-alphabetical language though, is the sheer number of characters required to communicate. J. S. Edgren describes the 'two major printing projects sponsored by the imperial court in the eighteenth century produced two founts, each of around 250,000 types, one of bronze and the other of wood'. This is a huge investment for a printer, and somewhat changes the trajectory of the printing industry compared to printing in other scripts. The last thing to note here is that Edgren briefly comments that Chinese printing is thought not to have taken the same trajectory as it did in the West because of the lack of influence of capitalism, changing the way in which texts were produced and distributed. ^1

Peter Kornicki describes Korea as holding what is widely considered to be the 'oldest datable printed item in the world' (a Buddhist text from 751) and the earliest extant example of typography, coming a hundred years before Gutenberg. Early printed texts were largely produced for ritual purposes, not for reading, but by 1010, work had already begun for a Korean edition of the Buddhist canon as texts for 'study and devotion' became more commonplace. The oldest extant book printed using metal movable type rather than wood blocks comes from Korea (1377), but evidence of metal type can be dated to the twelfth century, making Korea not merely adopters of this Chinese technology, but innovators in it. ^2 Cultural factors still prevent the kind of explosion of printing that we see in Europe - both due to the types of texts being produced and their availability, and the fact that Korea was under invasion during some of these crucial innovations.

Xylography reaches Japan in the mid-eighth century, but printing only really takes root in a few specific areas of written work. Most printed texts were Buddhist texts written in Chinese, with some secular printing taking place for medical texts, and Confucian works. Kornicki argues that classic works of Japanese literature were not printed until the seventeenth century, probably because the court society which had produced and transmitted them remained hermetic; another factor was the close association between calligraphy and literature which printing would have threatened'. Kornicki summarises to say that 'there is until the seventeenth century no trace of a book trade or of the kind of commercial printing that had already established itself in China'. Japan is another culture with access to the printing press far earlier than western Europe, but hopefully as this shows, certain conditions have to be met before printing becomes the kind of practice for which a culture becomes known. ^2

So what exactly did Gutenberg do, and why do we know him as the father of printing?

Gutenberg's printing press itself is not a novel invention entirely of his own - rather it is an adapted screw-press, which was a technology that had already been in use for some time in pressing fruit. However, this type of press made production reasonably efficient, and capable (eventually) of meeting commercial demand. As we have seen, Gutenberg isn't the first to use movable type either, but crucially, he is the first person in Europe to do so, and the first do come up with a means of manufacture for the type which was commercially viable (again... eventually) and which produced consistent results. In A Guide to Early Printed Books and Manuscripts, Mark Bland discusses how 'The importance of Gutenberg's original invention was not that he created the press, but that he found a way to manufacture movable type: that is letters that were exactly the same as each other and that could be reused once they had been distributed back into the case.' The process which Gutenberg used involved cutting a steel punch in the shape of the desired letter about 45mm long (steel is a hard metal) and hammering this into copper (which is soft). This copper matrix was affixed into a handheld mould and a mixture of molten lead, tin, and antimony was poured in. The copper matrix ensured the consistency of the figures. If the piece was poorly-cast, it was simply melted back down. Successful casts were filed down to a uniform height (about 24-27.5mm). Producing the steel punch was a skilled job (originally goldsmiths filled the role, though it later becomes its own craft). ^3

The ability to consistently produce type on a commercial scale is a train set in motion by Gutenberg, and conditions in Europe allowed print to flourish. Printing the Gutenberg Bible was a rocky enterprise which left Gutenberg bankrupt - it took a long time before printing was anything other than a risky business enterprise. Type was still very expensive, and was often inherited and passed down from one printer to another (see Blank, p.130 for some examples of this taking place). The big, BIG difference here is that the Roman alphabet needs only 26 letters and a few items of punctuation to function. It is a substantially more achievable enterprise to produce texts in an alphabetical language with comparatively few characters than it is to produce a text of any size in say, Chinese.

Paper was expensive and had to be bought in bulk before printing could commence, making a print run a big investment. Most printers in England at least, did not stay in business long. Nonetheless, a shared scholarly and religious language (Latin), well-established trade links and a (broadly...) capitalist economy gave printed books the impetus they needed to take off, and markets for printed books diversified quite quickly. This really interesting article by A.S.G. Edwards looks at the ways, for example, that William Caxton (who brought printing to England) quickly sought to diversify the book-buying market. Printing in the East remained (again, broadly) remained more insular, and in many cases, focused on prestige texts for a rarefied market for longer.

It is definitely problematic that Western sources tend not to acknowledge the far earlier work in printing undertaken in China, Korea, Japan and Vietnam, and as I hope I have shown, each of these cultures produced exciting and beautiful technologies for the production of texts. But besides the casual Eurocentrism of most discourse around this subject, your answer to 'how come the Gutenberg Bible is widely credited as the first printed book?' at least in the popular imagination, it has a lot to do with the explosive way in which Gutenberg's technology took hold in Europe and in a relatively short space of time, offered immense new opportunities for the transmission of information which were quickly taken up.

I hope others can step in and offer more interesting details - this has been an incredible 'broad brush' style answer, but this hopefully addresses the crux of the question with plenty of room for other, more knowledgeable folks than me to fill in the (many) gaps!

^1 J. S. Edgren, 'China' in A Companion to the History of the Book ed. by Simon Eliot and Jonathan Rose (Oxford: Wiley-Blackwell, 2009), pp. 97-110.

^2 Peter Kornicki, 'Japan, Korea and Vietnam' in A Companion to the History of the Book ed. by Simon Eliot and Jonathan Rose (Oxford: Wiley-Blackwell, 2009), pp. 111-125.

^3 Mark Bland, A Guide to Early Printed Books and Manuscripts (Oxford: Wiley-BLackwell, 2010), pp.130-131.