In appendix A.2 of the orange book, Ravenel defines a ring spectrum $E$ to be flat if $E\wedge E$ is equivalent to a coproduct of suspensions of $E$. (Call this definition (1).) I've seen this definition used in talks and the like as well, but I'm confused about where it comes from. I know two meanings of the word "flat" in this context. Firstly, Lurie's version (2): a spectrum $E$ is flat (over the sphere spectrum) if $\pi_0E$ is flat as an abelian group and $\pi_0E\times\pi_n\mathbb{S}\to\pi_nE$ is an isomorphism for all $n$. Second, the version defined in e.g. the nLab page on the Adams spectral sequence (3): a ring spectrum $E$ is flat if $E_*E$ is flat over $\pi_*E$.
As far as I can tell, these three definitions are all different, though I guess (1) implies (3). What I would like to know is, why did Ravenel call this condition "flat", and what is its precise relationship to the other two definitions?
