No it doesn't. A 6-20 outlet is only 3 wires, since it just a 240V outlet. It is just the two hots and a ground--no neutral wire. So it is exactly the same three wires that a wall connector would use.
I see this frequently, where some people think the term "NEMA" always refers to one specific type of outlet, and even more specifically, to the 14-50. But "NEMA" is like the word "metric". It's the name for the overall system. So a NEMA 6-20 or a NEMA 14-50 are different things with different wiring requirements.
NEMA connector - Wikipedia