I have a few questions relating to the BBD decomposition theorem.

I have come across the following two versions of the decomposition theorem.

Version 1.Let $f : X \to Y$ be a proper map of complex algebraic varieties and suppose that $X$ is smooth. Let $IC_X = \mathbb{C}_X[d_X]$ be the "constant perverse sheaf" on $X$. Then the derived direct image $f_*IC_X$ is a (finite) direct sum of simple perverse sheaves on $Y$, i.e., a direct sum of shifted intersection cohomology complexes $IC(Z,\chi)[i]$, where $Z$ ranges over irreducible closed subvarieties of $X$ and $\chi$ over irreducible local systems on a Zariski open subset of the regular part of $Z$.

There is also a different version.

Version 2.Let $f: X \to Y$ be a proper map of complex algebraic varieties. Let $K$ be a constructible complex which is semisimple of geometric origin. Then $f_*K$ is also semisimple of geometric origin.

I understand that the second version is more general than the first. What puzzles me a little is that the first version of the theorem does not mention perverse sheaves of geometric origin at all. I know that if $X$ is irreducible or smooth then $IC_X$ is in fact semisimple of geometric origin. Now version 1 seems to suggest that, in the setting of that version of the theorem, every simple perverse sheaf on $Y$ is of geometric origin. Is this true? Or are some simple perverse sheaves not of geometric origin, so they cannot occur in the decomposition, but version 1 of the theorem fails to mention this fact? Or does one perhaps need to impose some stronger assumptions on the variety $Y$, like projectivity? I would also appreciate any further comments/insight about how these two versions of the theorem relate to each other and precisely what assumptions one needs to make for these theorems to hold.

There is also a refinement of the first version of the theorem.

Version 1.a.In the setting of Version 1, suppose that there exists an algebraic stratification of $Y = \sqcup_\lambda S_\lambda$ such that $f : f^{-1}(S_\lambda) \to S_\lambda$ is a locally trivial topological fibration. Then the direct summands in the decomposition of $f_*IC_X$ are of the form $IC(\overline{S_\lambda},\chi)$, where $\chi$ is an irreducible local system on the stratum $S_\lambda$.

Question: is there an easy way to see how one can deduce Version 1.a from Version 1?