I was thinking: filter X as a colimit of finite spectra X_n such that the first stage is F. Then all other stages have their identity map factoring through F so theyre summands of F, so X is also.

You could also argue via the fiber, which would be a finite F-->X through which all maps finite-->X factor. I think that forces X to be finite already?

Also, your example pair is a counter-example, since those two spaces don't have the same rational stable homotopy groups (=rational homology)

Why does (3) require highly-structured associativity? That definition appears in Adams blue book from the 70s which, it was my understanding, doesn't deal with any high structure

Is this equivalent to definition (3)? One direction seems to follow from pi_* commuting with filtered colims. For the other direction if D is the filtered diagram of free E_-modules whose colimit is E_*E, then E_*( )(x)_{E_}D is a diagram of homology theories which I think lifts to a diagram of spectra (the nodes are free E-modules) whose colimit has a map to E(x)E by universality, and by another pi_*-commuting with filtered colims that map is an equivalence I think.

@JeremyHahn I hope not, because it's exactly such an E_infty map I seem to have constructed!

@FernandoMuro 10 bucks? Deal! On a more serious note, can you expand? (I think I have a proof of the positive answer, very much need to check the details)

Isn't the Lie bracket (Whitehead bracket) present before rationalization?