Once it's on the FPGA, it is certainly not a bitstream. The bits all end up scattered around in different internal flipflops and SRAM blocks. There is no stream anymore. So it's a "configuration" now, not a bitstream. A bitstream is just that... a stream of bits. In this case, the bitstream encodes the FPGA configuration.
Firmware is a more abstract concept. Firmware is the stuff you have to apply to the device or it doesn't work. I hesitate to provide a more specific definition, since more specificity would undermine the usefulness of the term. Firmware doesn't have to be a single bitstream, though it often is. It could be multiple, or it could refer to some other way to structure the data. Consider multiple files in a typical computer filesystem. These files could, together, represent the firmware, but they have tree-type relationship and aren't just a single stream (a tree of streams, of course).
So if all a system needs to "work" (whatever that means) is an FPGA configuration bitstream, then it represents the entirety of the firmware. Otherwise there may be other files or bistreams or whatever you want to say, and those all together represent the firmware.
Addendum: it's worth noting that the firmware doesn't have to be compiled, whereas a bitstream (like the software term, a "binary") always is compiled and ready to run or be applied to the device.