"The Author warrants that the Work has not been published before in any form except as a preprint."
The distinguishing characteristic of a preprint here is that it has not been reviewed. See also the SHERPA/RoMEO definition of a pre-print.
Academic publishers want to be the first to publish a paper and give it the 'seal of approval' that it has passed review.
So what this publisher is essentially requiring is that the paper was not reviewed and accepted for publication elsewhere.
If all you did was posting your pre-print on Arxiv (or some other repository/site), you comply with the publisher's wish.