This was easy to find out. It was a matter of trials and errors. It does not matter whether an m-file is actually a function or a script if it will go classified into Functions or Examples in the File Exchange site. It simply depends on whether an html-file was generated or not using Publish > Publish. Those m-files with html-files Published whether as a "User Default" or as a "Functions" will be classified as Examples, and those without html-files as Functions.
This functinality seems to be a bug in the FileExchange site, as often times one might want to have a well-formatted html-files for help for function m-files.