Did you UNIQUELY NAME the Hard-drive (IE not Untitled or the like)?
There are no other drives of a similar name?
Is this drive mac formatted? (if so, bingo-there’s your problem; permissions on the drive, set systems to ignore it). This might solve your problem for your next project: Format the external as fat32 or exfat. No permissions issues. HOWEVER…
Cacheing should be unique to each system (it is not necessarily Project specific though it does have a note in the project file). The cache is the description of active memory and should be rebuilt each time. But it can hold a multitude of data from different projects. Deviating from the standard location just makes it have to search for and load the cache every time, into active memory, and that takes a lot of time to balance.
The preview files are what conform for playback. Audio conforms… …Make sure your audio input matches your sequence. If previews are intact, it should playback very quick, with the previews intact. Leave the settings for cacheing alone. You need a big enough main drive to hold the cache, but you can dump the cache at any time without hurting your project. You are better off leaving the caching on your internal drive (leave it alone).