Which directory to save .fzpz and use as my own part?
It does not really matter where it is saved. Once you have fritzing running you go to the parts selection area (bins) and in the top right corner there is a little drop down menu with an import function. Once it is imported it will be in your “my parts” bin.
While it is true Fritzing uses the mine (and for sketches, the temp bin) for parts by default (and the location for that by operating system is available in the parts format document on github.) I find it a good practice to use the export function (right click on the part in the bin and click export part) to save a copy of the part in a directory not controlled by Fritzing. You can use any directory name you like (I use fritzing_backup). It is possible to corrupt the parts database and need to clear it which will clear the parts in the mine parts bin.