It doesn't imply that it's the disk operating system so much as it implies that it's the disk-operating system. You could boot an Apple II from ROM, enter and run BASIC programs, load programs from cassette, and basically do whatever an Apple II can do, but without DOS there was no way to access files on disk. Apple DOS didn't really do any of the features of a modern "operating system" (which were either built into ROM or entirely nonexistent), but what it did do was provide the routines for accessing disk files and directories, thus disk operating system. The same can be said of Atari DOS, CBM DOS, TRSDOS, etc.
Although MS-DOS isn't entirely the same, to some extent it is. The earliest IBM PCs did come with BASIC in ROM (called "IBM Personal Computer Basic" on the copyright screen, but commonly known as "Cassette Basic" to differentiate it from "Disk BASIC" which ran on DOS), and the BIOS ROM provided the basic facilities of keyboard, screen, printer, etc. So the most important service that MS-DOS added to that was the ability to read and write files from disk, and execute programs on disk. It also added a command shell (COMMAND.COM) that was the most common way of interacting with the computer, rather than BASIC. When later PC models eliminated ROM BASIC, DOS became pretty much the only way to boot and use the machine at all (unlike most other 8-bit machines).