What makes a publisher traditional varies a lot. It comes down to editing, design, promotion, and business. A publisher that doesn't edit a book has a strikingly different idea of what traditional means. It doesn't mean the book will be un-edited, but bringing your own editing to a publisher, without their review and revision, is a good deal short of the business practices for a publisher known as traditional.
Design from the ground up, starting with graphics choices and through typography and interior layout, is also a bedrock deliverable in a publishing contract. You can always help out with images, but it's the publisher who's responsible for selling the book. They will want a cover that will sell in the postage stamp-sized world of Amazon and Apple Books and Kobo and Barnes & Noble.
You don't even need a traditional printer to call yourself a publisher. Lots of digital-only titles are available, from midsize to bigger publishers. Lots of them are in the genre ecosystems. Many very small publishers, though, ensure you get printed. The hybrids even offer a number of author copies in the packages they sell. Those are books that cost about $4 each to print and ship, so you can decide if those copies are a real value item in a hybrid's package.
We're into the shifting sands once we consider promotion and business. Who's going to list your book, where it can be ordered, its discounting, accepting returns, creating sales page material — all business. Promotion is called marketing in some quarters, and if you're lucky your publisher has experienced resources to help your book get discovered and recommended. A traditional publisher has a better shot at getting a review from Booklist, Library Journal, and Publisher's Weekly. A traditional will create a one-page promo file for reviewers and journalists to consider.
A great publisher has a sales force. These are the reps who contact buyers. For example, that's someone like Robert Poole, Costco's book buyer. Poole probably has a few staffers, but these are the people who get pitched by a publisher's sales reps. Or the reps pitch to a distributor, who then gives the publisher better play in the catalog and their sales pitches to people like Poole.
We all get the best deal we can. Smaller publishers are good too, and we understand that we'll have to shoulder some of the promo (and publicity!) if our publisher is smaller. At a minimum, though, they should give us a professional edit, list our books, and get us a good-selling cover. They risk the hours their pros must work in exchange for the share of the sale the publisher takes. Publishing is all about risk, and who will bear it. That's what makes it a business.