I think it is fair to say that Disk Operating System implies an OS that talks to disk in the commonly perceived environment of files, file sizes, directory(ies), time stamps, open(), close(), read(), write(), etc. Operating System is neutral on the subject of whether or not there is a disk. Many might group them with devices that have no large, native storage, such as many classic PDAs, or that used cassette as the primary storage medium. As the question implied, we have sort of moved beyond the idea of DOS. Yes, Windows, OS X, and Linux can all talk to disk (HDD and/or SSD), but in the networked world, that is a feature we no longer really think about.
But something else exists. Classic FORTH broke all the perceived rules. Most implementations had access to disk. But no one referred to it as a DOS. Heck, most people would not even recognize it as an OS, even if it allowed a full, interactive, multi-user, (co-operative) multi-tasking environment in 64 kbytes of RAM with an 8-bit processor. Even if the OS was indistinguishable from the language. Or the application(s). Or the development environment. You had the hardware and you had FORTH, and that's all there was. If you were operating single-user, you had 100% control of everything, all the time.
The thing is, even without files, directories, or time stamps, it was an OS that gave R/W access to disk. At a high level, just four FORTH words (keywords, functions, whatever term you favor) were used to interact with the disk
- BLOCK (n .. addr): given the block number n, read it into memory (if not already present in one of the available buffers) and return the address of the first byte.
- UPDATE : mark the most recently used BLOCK as needing to be written back to disk before it is over-written
- SAVE-BUFFERS : immediately write all updated blocks (if any) back to disk
- FLUSH : perform SAVE-BUFFERS and force re-read of any block, even if it is still present in memory
There were other words too, both higher and lower level, but those were at the heart of this operating system thingy that talked to disks.
I'll add that most modern FORTHs cast away their operating system nature and operate as simply one application among many under the "real" OS.