There is no formal definition of "operating system". Some used to maintain that "operating system" and "file management API" were one and the same, with the OS having nothing else to do other than provide a command analyzer. (After all, this is all that MS-DOS did, originally.)
I've always maintained that DOS wasn't a real operating system -- that an operating system's job is to abstract and virtualize the hardware, and to manage hardware resources. DOS did essentially none of that.
As to whether a file system is a part of the OS or a part of the "storage device", a lot depends in turn on what you mean by "file system". There is the physical layout, such as the layout on a floppy disk or CD, and there is the file system FUNCTION, which depends on having some intelligent entity (CPU or peripheral processor of some sort) to take the nonsense on the disk and return it as a meaningful sequence of bytes. The layout presumably conforms to some standard, so you can, eg, record a CD on one device and read/play it on another. The question is whether this layout is a "file system", or whether the "system" resides in the devices clever enough to read/write the layout.
In most computer contexts, one uses the term "file system" to refer to the APIs that allow you to read/write files, and the combination of CPU and peripheral devices, operating under the control of some OS, that implement those APIs -- the term does not usually refer to the physical format of the media, or individual media, whether removable or not.