My two cents:
PD stores the audio clip files in c:\Users\Public\Cyberlink\Downloaded Audio\
In that directory you will find a number of folders. It looks like a folder is created as soon as you play a sample and/or downloaded the clip. In each of the folders there is a .m4a file (with the tag preview in the file name), and an .mp3 file (after you have downloaded the clip).
So, PD will obviously download the files in order to play them, but thereafter apparently stores them for later. You will need an internet connection, (and the Cyberlink server will have to respond), and you need to have access to the folders in User\public…. As in the above. My assumption is that when there is no connection and/or no download because of any other type of error occurs including not being able to access or create a folder, the “Not connected to internet” message is displayed.
I would also assume that at PD installation time AND at installation of any of the audio (video etc) Packs via the Application Manager the above-mentioned folders will be accessed too. So, there were either error messages or something with the folders and/or folder access may have happened after the installation and the moment of the editing of your other video.
Maybe this helps finding any differences between your 2 systems (you say you are running PD on 2 different but identical systems?) that may cause the No Internet connection message.