Who is the publisher - me or Lulu?

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I publish all my books with Lulu. It is free to publish with them, if I do everything myself to publish a book, which is exactly what I do. I don't pay someone to publish books for me. (They do have those services though.) I format the manuscript, I edit it, I create the cover, and I actually go through their entire publishing process to publish a book. Technically that makes me the publisher, not Lulu, because I do all the work. But I always put in the copyright page Published by Lulu Press (NOT Published by Lena Kovadlo); and in the publisher field on Lulu I say Lulu not me. I do this because I use a Lulu-owned ISBN. They say in the step of the publishing wizard, where you get the ISBN for the book, that if you use their ISBN they act as publisher. Even on Amazon, and in other online retailers, it says lulu.com in the publisher field for all of my books. The dilemma is that in their publishing guide, and in the final step of the publishing wizard, where you enter the publisher name, it says Lulu is not the publisher, I am. This makes me wonder if putting in the copyright page of all my books Published by Lulu Press was incorrect and that I should have instead written Published by Lena Kovadlo. Any ideas??? I wonder though... would the publisher field in online retailers' websites would still say lulu.com even if on Lulu I'd enter my name in the publisher field?