Searching and starting from Noam's answer, I found some extra information:
the phenomenon pointed out by Noam on automorphism groups of extraspecial 2-groups seems to appear in: R.L. Griess, Jr. Automorphisms of extra special groups and nonvanishing degree 2 cohomology. Pacific J. Math 48 (73) 403-422. (Link, MR link)
The discussion about $d_1$ and $d_2$ seems to appear in: W. Feit and J. Tits, Projective representations of minimum degree of group extensions, Canad. J. Math. 30 (1978) 1092-1102, and continues in P. Kleidman and M. Liebeck. On a theorem of Feit and Tits, Proc. AMS 107(2), 1989, 315-322. (Link, MR link)
Feit-Tits establish that if $d_2(S)<d_1(S)$, then $S$ is of Lie type over a finite field of characteristic 2, and in this case $d_2$ is a power of 2. Kleidman and Liebeck establish the precise list of those such groups of Lie type that indeed satisfy $d_2<d_1$, including the computation of $d_2$.
In particular, it follows from this classification that the smallest dimension in which some finite simple groups ``occur" but not through a projective representation is 16, namely for the groups $\Omega_8^-(2)$, $\mathrm{Sp}_8(2)$, $\mathrm{Sp}_4(4)$.
Besides, if we consider representations over an algebraically closed field of characteristic $p\ge 5$, defining $d_i^{(p)}$ in the same fashion: the picture is exactly the same. In char. 3 it's essentially the same up to a minor difference in the classification among those $S$ of Lie type over a field of characteristic 2. On the other hand in characteristic 2 it's dramatically different: $d_1^{(2)}(S)=d_2^{(2)}(S)$ for all $S$.