The DOS for the Commodore computers including the above mentioned Commodore 64 was not an extension to BASIC. The DOS for Commodore computers instead ran on the floppy drive units themselves which included their own RAM and 6502 family CPUs to do so.
The other 8-bit micro I used personally was the TRS-80 Model 1. It would boot directly to BASIC but their was no DOS as part of this basic. To use a floppy drive a DOS boot disk was required. There were a few different DOS's available for the machine. The official DOS available from Tandy was TRS-DOS.
Regardless of where it ran, to the C64 user, at a command level, it looked and acted like an extension to BASIC. You can see some examples of the basic code here: https://en.wikipedia.org/wiki/Commodore_DOS
The other 8-bit micro I used personally was the TRS-80 Model 1. It would boot directly to BASIC but their was no DOS as part of this basic. To use a floppy drive a DOS boot disk was required. There were a few different DOS's available for the machine. The official DOS available from Tandy was TRS-DOS.