What I've seen driving the definitions is the gate count of the processors and thus the power consumption.
Embedded processors have been simpler. They lack performance-tweaking extras like integer divide instructions, branch prediction logic, floating-point instructions or instruction/data caches. They are aimed at running smaller fixed-in-place programs. Using fewer gates means:
* More space on-chip for peripherals and memory, for increased functions in a microcontroller or FPGA.
* Chip can be manufactured in an older, larger geometry silicon process, making it cheaper.
Application processors use more gates to put the performance-enhancing stuff in described above. They are aimed at running larger programs. Might also have an mmu so it can run a protected os, multi-tasking etc.
If memory serves, Altera NIOS II and later ARM Cortex processors look at it this way.