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).

In `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.)