With Raspberry Pi is the Linux OS running in the firmware (like an OS on a cell phone) or on top of a firmware (like on a regular PC)?
I do not understand the question.
The only firmware the Pi has is a bootloader which is responsible for reading the boot sector from the SD card.
All software (including Linux and applications software) is then loaded and run from the SD card.
If you think of the SD card as a flash drive you could liken this to a cellphone. If you think of the SD card as a hard disk drive then you could liken this to a regular PC.