The short answer is that you’d expect
lib to be built (or rather, installed) if there’s a library to install, and
include if there are headers to install. Usually the two go together (at least for C and C++ libraries).
binutils’ case, the library and associated headers are
libbfd, which is a library used to manipulate object files in a variety of formats. It is installed by default for same-host compilation, but not for cross-compilation, which is why you’re seeing different behaviour in your scenarios. You can see the conditional default in
bfd/acinclude.m4 in the source code.
(You only need
libbfd if you want to build GDB.)