Is the name of the tar file you downloaded a typo (3.2.0) ?
No, in fact the installer version is 3.2.0 in SourceForge: http://sourceforge.net/projects/freepascal/files/Linux/3.2.2
.... and the file located there is named "fpc-3.2.2.x86_64-linux.tar" and not " fpc-3.2.0-x86_64-linux.tar"
These are two completely different tar files.
Like this?
No, not exactly

You need to locate where you installed the unit files (you have installed FPC on your system and I have no clue what location you used to install the complementary unit files). The location where those files are located is named a path (not to confuse with a system search path that is used for searching directories in order to locate executable files).
Do I need to edit the path?
The -Fu paths located in the fpc.cfg file, yes in case that location does not match the location where your units files are actually located. You do not need to replace the version number and target macro's, only point -Fu to the correct location (path) where your units are located (where you have installed them to).
But the point of that remark was that if you actually used the 3.2.0 tar archive and also have FPC 3.2.2 installed that things are becoming a bit messy. It is currently not even clear if you installed 3.2.0 and 3.2.2 together (one using the package manager of your distribution and the other using the tar archive and installed into your home directory) because something like that is not going to work.
edit: and I am a bit slow as user tetrastes already figured that so please follow his suggestion.