They actually describe it as "6+1 degrees of freedom" [1] with the gripper being the "+1" - so it's got base, shoulder, elbow, wrist1, wrist2, wrist3, and gripper.
This is a conventional way of describing things. Traditionally robot arms come with a "tool flange" where you attach your own "end effector" (which might be a gripper, or a suction cup, or a welding gun, or a paint sprayer, or whatever) and we count the degrees of freedom before the tool flange separately from those after the tool flange.
Occasionally robots come with 7 degrees of freedom [2] which gives you more options for reaching the same tool flange position. This can be useful in certain applications, like working around obstacles in the environment. It's uncommon though.
It should be noted that 6 degrees of freedom is the minimum that enables reaching any point with any orientation of the effector, while 7 degrees of freedom is what a human arm has with the shoulder fixed, which enables reaching some points even around some obstacles that would block a 6 degrees of freedom arm.
Including the movements of the shoulder, a human arm has 9 degrees of freedom, but the additional 2 degrees of freedom do not provide a new capability, they just extend the range of possible motions beyond that limited by the joints of the other degrees of freedom.
An arm with many degrees of freedom, like an octopus arm, could reach some places even when having to avoid many obstacles.
His point is that the robot has split the kinematics to have 3 DOF for position near the base and 3 orientation DOF on the wrist.
This still gives you 6 DOF on the end effector, which is pretty good, but overall the arm design is restricted in its ability to route around obstacles.
This downside has an upside though. Since every cartesian position has exactly one pose for the first 3 DOF, the inverse kinematics are simpler and you do not run into singularities for basic position control when maintaining a constant orientation.
I'm sorry, I was confused. This arm appear to have wrist pitch and yaw "backwards" from standard split forearm configuration, and I mistook the wrist yaw motor to be something else.
Although, it does still worry me that there don't seem to be a lot of footage of this arm with that axis away from its neutral position...
so, six variables that produce a posture. 6DoF.
but explaining this makes me feel like i'm missing some deeper meaning in your comment?