What is the complete, authoritative way in which Mac OSX 10.9 and later builds its PATH?
You're trying to understand the theory and that's great.  I'll just say that from practical experience, you can modify .bashrc, .profile and/or .bash_profile and the path setting will stick.  I prefer ~/.profile simply out of habit.  My teammates all use .bash_profile.  Either way it works as long as you pay attention to the order of processing precedence.

Here are a few other thoughts on the matter:


Homebrew wants me to amend my PATH (stackoverflow)
How to correctly add a path to PATH (stackexchange)


Hope that helps.