Chope it's the other way around really.
The Public version needs to be installed in the publicly visible directory of your site. Most users probably need this approach as they dont know what a public directory is or they use a subdirectory for installation.
The non public variant is the safest, according to security recommendations. You install the files one directory above the public directory and then copy everything from Public over to the publicly visible directory, like public_html.
Using composer is the most flexible solution, but indeed requires some perseverance.
