How -exactly- do I remove a specific directory, from my PATH, in OSX, Mavericks?
It depends where the PATH is being set. If you use bash, the path could be set in any of the following:

/etc/profile
/etc/bashrc
~/.bash_profile
~/.bash_login
~/.profile
~/.bashrc


This list is in the order that the files are read (by bash). Later files override earlier files.

Check the above in reverse order for export PATH which will write your PATH variable.

You should be able to find where it's set by launching your shell with -x.

Also check /etc/paths and /etc/paths.d for paths that are set before your shell is started.

If all else fails, you can set the PATH yourself to override its current contents. Run echo $PATH, then paste your PATH after export PATH in your ~/.bashrc and make your desired modifications. As ~/.bashrc is the last one read, it will override any earlier modifications to your PATH.

If you use zsh, it's probably in ~/.zshrc or a sourced file.