> there's a difference in the scripts depending from where they are called.
There are two scripts, each for a different purpose.
> Is there any log-file where I can check the chron activities (in addition to 'Site administration/ Notifications')?
Since you call the web-based cron, it will be logged in the web server's logs. Your hosting provider should know.
> Having some trouble with my hosting provider around cron jobs sometimes and the whole system is quite intransparent for me.
That is why people go to providers who specialize on Moodle!