What does the author intend by operating system directory structure?

I'm reading Avi Silberschatz's (9th Edition) Operating System Concepts, and in section 11.4 File-System Mounting, the author discusses the phases of filesystem mounting as follows:

• The name of the device and the mount point—the position inside the file structure where the file system is to be attached—are supplied to the operating system.
• The operating system then checks to see if the device has a proper file system.
• Finally, the operating system records that a file system is mounted at the given mount point in its directory structure.

I'm perplexed by the final step since, to the best of my knowledge, the directory structure is saved someplace on the disc and contains information about the files such as name, location, size, and type. So, ¿what does the author mean by operating system directory structure? ¿Is it the same disc directory?

Furthermore, ¿which phase completes the transfer from file name to physical disc address? Is it done by the disc driver, the disc controller, ¿or the CPU with memory?