Let $X$ be a spectrum and $E$ another spectrum (it'll be our coefficients, if it makes things easier I'm ok woth assuming $E=H\mathbb Z$)
The definition of $E_nX$ is usually given as $\pi_n(E\wedge X)$, but if you pick a specific model for spectra (e.g. symmetric spectra) there's another natural construction : consider $E_{n+i}(X_i) \cong E_{n+i+1}(\Sigma X) \to E_{n+i+1}(X_{i+1})$ and take the colimit of those maps along $i$.
(Note : in this second construction, $E_*(Y)$ for a space $Y$ is defined via the first construction, with $\Sigma^\infty Y$, which makes it easy to see that if $X$ above is $\Sigma^\infty Y$ for some $Y$, then they do coincide)
It feels like they should coincide, but at this point I know too little about spectra to prove that they do, so that's essentially my question : are they the same, and if so how can one prove that ?
I thought of proving it by expressing $X$ as some sort of homotopy colimit of the $\Sigma^\infty X_i$ but I'm not sure that's going to get me anywhere
The answer is yes, they are the same whenever $X$ is a semi-stable symmetric spectrum (so it will be yes, up to replacing $X$ by a nicer spectrum); the proof is in Schwede's Symmetric spectra, Prop. II.6.5.; where there's also an analog for cohomology.
The idea is essentially to write $X$ as a homotopy colimit of the $\Sigma^{-n}\Sigma^\infty X_n$, and to use the fact that $E$-homology behaves well with respect to homotopy colimits (because $\mathbb S$ is compact in the stable homotopy category)