Let $E$ be a (pre)spectrum (in the most classical sense, i. e. the sequence of CW complexes $E_n$ and maps $SE_n \to E_{n+1}$). Then we have the generalized cohomology theory $E^*$. For finite CW complex $X$ we have that $E^n(X)$ (reduced, as on the corresponding suspension spectrum) is equal to the direct limit $\lim_k [S^k X, E_{n+k}]$. Is this true for all CW complexes? I think, no. But I've seen this definition of the cohomology theory corresponding to spectrum in some books. Is it just inaccuracy, or maybe it's different non equivalent definition, is it really cohomology theory, is it exact? And maybe this is true for some nice prespectra (not for $\Omega$-spectra, of course. I'm mainly interested in the case of classic Thom prespectra).

In general, no. Assuming, say, that the structure maps $SE_n\to E_{n+1}$ are inclusions, the correct statement is $$ E^n(X) \cong [X,\lim_k \Omega^k E_{n+k}], $$ and if $X$ is not compact that is not the same as $\lim_k [X,\Omega^k E_{n+k}] = \lim_k[S^kX,E_{n+k}]$. Peter May drilled this into me when I was a student (and after!). The general statement (for $E^0(X)$) is the adjunction $$ [\Sigma^\infty X, E] \cong [X, \Omega^\infty E], $$ comparing maps of spectra on the left to maps of spaces on the right. For nice $E$, $\Omega^\infty E = \lim_k \Omega^k E_k$.