New term for "disk"
A very generic term would be storage media, which I remember being used to describe all sorts of data storage devices from over 25 years ago. However, it applies to CDs and DVDs (both of which are actual disks) and other removable media in addition to less mobile versions such as internal drives and network attached storage.1
If you want a term specific to what a typical standalone digital device would use, you can refer to its internal storage 2, or perhaps more succinctly storage.3
- Forensicon
- PC.net
- Wikipedia
Your addendum has changed the nature of the question. You are not looking for a term to describe the physical device from which the data is written and read. Instead, you want a term to describe the service an operating system provides to store and retrieve files. The file system (also filesystem) is that service.
Without a file system, information placed in a storage area would be one large body of data with no way to tell where one piece of information stops and the next begins. By separating the data into individual pieces, and giving each piece a name, the information is easily separated and identified. Taking its name from the way paper-based information systems are named, each group of data is called a "file". The structure and logic rules used to manage the groups of information and their names is called a "file system".
Wikipedia
Computing terminology is full of anachronisms. See the picture below. It comes from my Microsoft Word 2013 edition.
That is a perfect picture of a floppy disk! You won't see one of those outside of a museum now.
I think that a lot of the time we just pick up a word and its meaning without thinking about it. Eventually someone comes up with the ideal term and we just switch overnight.
Example - In Britain for many years we didn't know what to call the thing that changes the channel on your TV. It had various unsatisfactory names, e.g. 'zapper'. One day someone called it a 'remote' and now everyone does.
Because technology moves so fast, I don't think it matters if popular terminology fails to keep up.
If I want to buy a Solid State Drive, I go into a shop and ask for a SSD. However if I asked for an SSD drive or even an SSD disk, I'm sure they would still take my money.