Here's the battle of the century of r0ach vs smooth regarding this issue. They call him "smooth" because it's like talking to Bill Clinton. You tell me who won:
I suspect you are attempting to justify other consensus mechanisms by trying to find loopholes in definitions to prove a point. Satoshi did solve the byzantine problem in the face of sybil attack, it's been proven.
There have been other solutions, such as ripple's consensus - which is only resistant against 20% byzantine failures. By applying proof of work to the problem, satoshi was able to increase this to 50% which is the theoretical optimum.
Satoshi's solution is not perfect, of course - the pools centralise and ASICs worsen the situation, but the core of the idea is sound.