So in Romania the hybrid publisher behaves like a trade publisher except the writer pays for the publication of the book and the two entities split the profits from sales?
Based on the definitions supplied above, there's no difference between hybrid and vanity press except undefined "aspects of trade publishing" attributed to the former. "Hybrid publisher" appears to be "vanity press with a convincing spin."
The writer pays for the publication of the book (or sometimes they split costs) and if they split costs, they split profits too. The idea is that a small or medium publisher cannot take the financial risk of a book (investing everything in a book which fails to sell, in a very uncertain book market - our main problem here is the book market), therefore the writer (or his sponsors, because in some lucky cases, the writer has sponsors) has to share the financial risk.
I would not say that they behave like a trade publisher, because there are things they cannot do - usually not display your books in bookshops (except in some cases, in a few bookshops they have a contract with), they cannot help much with marketing because they have no specialised people for this, etc.
The difference between hybrid and vanity press is that a vanity press will print the number of books you ask for and give all of them to you to store and sell, with no marketing help at all. It is also a main difference that they will print what you paid for (without a vetting/ editing) because they live from your money.
A hybrid press is careful with its reputation and won't accept bad quality books. Some mediocre ones might sift, if they are salable and they do not affect negatively the press as concerns the subject, but usually they choose good ones. So vetting is ensured. They make sure a literary critic or a better known writer publishing with them writes the foreword and the fourth cover endorsement. Sometimes these publish book recommendations in literary magazines too. And when they print the books, they do not dump them all in your lap. They give you the number you asked for, and they sell the others through online bookshops and at literary events. Some of them also give you a public launching of the book. (Some cannot, usually because they are not in the same city, but at distance). But some of the ones being at distance still offer an online launching or book trailers and various kinds of media swag. And they help with the marketing too, with online posts and so.
I have never mentioned Ingram, nor Amazon, because they do not work here. If you want to self publish, usually, you take your book, already converted into a pdf, to a typography and they print it for you without a publishing house. Then you manage how to sell it yourself. Or there are a couple of newer print-by-demand platforms which do some services for you (not sure what, because I did not use them) and they sell your book on paper or e-book.
