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 $E$. 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: 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: 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?

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?