How do I distinguish between ‘binary’ and ‘text’ files?

You can use the file command. It does a bunch of tests on the file (man file) to decide if it’s binary or text. You can look at/borrow its source code if you need to do that from C.

file README
README: ASCII English text, with very long lines

file /bin/bash
/bin/bash: ELF 32-bit LSB executable, Intel 80386, version 1 (SYSV), for GNU/Linux 2.2.5, dynamically linked (uses shared libs), stripped