I believe the answer to your question (including the stronger version) is yes, and it may be written down explicitly in the finite group literature. Note that in the defining characteristic $p$ for a finite group of Lie type, the root structure plays a major role here since $p$-subgroups are the same thing as unipotent subgroups.
A basic source for the detailed structure of Sylow $p$-subgroups in all cases is Number 3 in the treatise The Classification of Finite Simple Groups by Gorenstein-Lyons-Solomon (AMS, 1998). See in particular their section 3.3, where for instance they have a convenient summary of the $p$-ranks of the finite groups of Lie type: this is the largest rank of an elementary abelian $p$-subgroup, important in the study of complexity and support varieties.
I'd have to look again at the literature including G-L-S to sort out what is actually known about conjugacy classes of elementary abelian $p$-subgroups, but in any case it's probably essential to relate the question to the Lie-theoretic structure inherited by the finite groups. In any case, the almost-simple groups are the essential ones to consider.