Eckert and Mauchly weren't reliable narrators of the history of their work, given how much they needed to claim in order to get the patents that were later invalidated. They claimed quite a few things that were later disproven in court or through historical study. At the least, Eniac was designed by the team at the Moore School.

von Neumann's "First Draft of a Report on the EDVAC" was both a summary of discussions by the Eniac team as well as von Neumann's own original work proving and expanding on various aspects of those discussions. It's called the von Neumann architecture because he was the author of the seminal paper, not because he laid any particular claim to the ideas. Eckert and Mauchly were mostly upset about the paper's existence, regardless of author, because it made it impossible for them to patent the stored program idea in general.