The short answer: whoever buys the ISBN from their national ISBN Agency. (In the US, that’s RR Bowker.) This is the minimum requirement for a publisher. Successful publishers do far more.
If you didn’t buy the ISBN from the Agency, you’re not the publisher, even if you pay a “self-publishing company” and make all the decisions. If you use one of their ISBNs, they’re the publishers, and it CAN matter. Why?
First: The ISBN identifies both the book and the publisher. The publisher will get all of the orders, information requests and returned books. It is possible to buy a single ISBN, but that doesn’t give a publisher ID. You’re better off buying a block of 10 or more.
And also: If the publisher is a vanity publisher, aka POD publisher, aka self-publishing company, your book will be judged by the quality of all of their other books. And that’s almost always bad.
Okay, so what else makes you a publisher? Do you need to do your own printing? No, but you do need to find and work with a printer. You would be wise to send out at least a dozen RFQs as well, before selecting one.
You also need to get your book’s cover and interior designed and laid out. You need to get the manuscript edited (copy and line editing are a minimum, and a deeper edit is usually a good investment). I cannot over-emphasize the importance of design and editing. There will be other posts on these topics later.
Publishers handle the distribution of the book. (Small publishers usually outsource this.) This includes soliciting orders from the large national accounts, getting the catalogs that include the book to buyers (not the end customer, the folks that order for the store’s stock) at the smaller bookstores, processing the orders, issuing credit and collecting on it, warehousing the stock, and picking, packing and shipping to fill the orders.
Publishers market the book. Authors are well-advised to do some of the marketing themselves, but publishers handle getting out review copies, getting blurbs, issuing press releases, giving appropriate and timely article ideas that might reflect upon the book, or draw attention to it or the author.
Publishers provide the capital to do all this. Authors get advances, which may NEVER earn out. The designers and editors do their work about 6 months before the book is published. (More or less, and what schedule does your company use?) Printers are paid before the books sell. Bookstores take 90 days to pay, if you’re lucky, after the books leave the warehouse. So the publisher puts up the cash long, long before there’s any possible return. And if the book tanks, and you have no sales, or sales and then 75% returns, the publisher still needs to come up with more money to finance the next book.
Did I miss something you think is important? Do you disagree with my definition?