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1. Forward
2. Backward
3. Left strafe
4. Right strafe
5. Upward
6. Downward
It more refers to the flexibility of movement than the possible states of position/rotation. In the game, we give the usual WASD for planar horizontal movement, Space/Shift for vertical, mouse X and Y for yaw and pitch, and QE for roll. Though you never actually need to roll to look somewhere specifically, it makes motion much smoother when performing vertical Uturns and such. In real life situations, an airplane that could not move freely over all rotational axes would put quite a bit of stress on its passengers and crew.
I think my weird misconcieved ramble stemmed from an explanation of why ColdEquation is correct even if you use something called Euler angles. My example of rotate right 90deg, pitch up 45deg, and then rotate left 90deg required three steps. The fact that it required three steps means that you have three rotational degrees of freedom, even if you try to describe your system with two Euler angles.
Silly me.
However try not to think of movement, but of positioning.
What I mean is how many variables are needed to describe an object's (current) position and orientation.
You need x,y,z for position and you need q,p,r for where it's pointing and how.
But even for what he says he's not totally correct. Degrees of freedom are "awarded" for motion "ability" not for ending position capabilities. If you had a robotic arm that could not do roll, but it could still reach all orientations by combining pitch and yaw, it would still be missing 1 degree of freedom.
After teacher intervention, engineering texts cited, and many thousands of bytes wasted on the Internet there remained a group of "sixth degree deniers". I guess I'm thankful for having played Elite all those years ago, since all you have there is Forward, Roll and Pitch, no Yaw or Slide. It made it obvious that you had six degrees of freedom in orientation and location even you only had three degrees of input freedom.
(and thus, using 2 degrees of rotation and 1 degree of translation, if I follow your theory, you can place anywhere you want with any rotation you want. In that cas, the two other translation degrees are not bound to be ?)
As Godeke said, if you think of it as a vector, a 3d solid object does not just have a vector indicating its direction its front is pointing; a third angle is required to specify which way the top is pointing.
That makes 6 ^^