I think your understanding is correct. I just wouldn't have put the additional E-quantifiers around E(Fb) and E(EG¬a).
Look at the path from s0 to s1, looping at s1. This is a path satisfying G¬a.
This path exists from s0.
Now look at the path from s0 to s2, looping at s2. It satisfies Fb from the first state. It also satisfies EG ¬a in the first state.
Similarly for S2. In s0, s1, there is a path satisfying G¬a. s2 is the only state that satisfies b. But as it doesn't satisfy EG¬a, there is no path satisfying F(b ∧ EG¬a).