I haven't heard of the term "primary electrode".
250.50 states that all of the electrodes as described in 250.52(A)(1) through (A)(7) that are present at each building shall be bonded together to form the grounding electrode system.
250.52(A)(1) is metal underground water pipe.
That means that the metal water pipe shall be an electrode.
250.53 (D)(3) Requires a supplemental electrode for an underground water pipe electrode. That can be any of the electrodes specified in 250.52(A)(2) through (A)(8)
250.68(C)(1) states that a connection to an underground water pipe electrode shall be within five feet of where the pipe enters the building. It further states that water pipe located more than five feet from the point of entrance shall not be used as a conductor to interconnect electrodes of the grounding electrode system.
You are correct that there is no definition of "primary electrode" in the code.
However, in my opinion, if the Ufer can stand alone as the grounding electrode for the house, then any other electrode would be an "auxiliary electrode" per 250.54. I would agree with you that the water pipe is an electrode, but since the Ufer could stand on it's own, the water pipe is "auxiliary".
250.54 specifically says that 250.50 and the supplemental requirements from 250.53 do not apply.
250.68 is about the connection of a GEC at the service, which, in my interpretation, means that if you were using the water pipe as your grounding electrode for your service, you would have to comply with 250.68.
However, if the water piping system is only an "auxiliary electrode", since you are using the Ufer, then 250.68 would not apply, because it only refers to the the GEC connection from the service to the grounding electrode. 250.68 would apply to your connection to the Ufer, but not to the connection of the water pipe.
So, the question is, is that water pipe an "auxiliary electrode" or a non-auxiliary electrode? If it is a non-auxiliary electrode, you are entirely correct. If it is an auxiliary electrode, then it only has to be bonded per 250.104 (A).
My only input on that is, what is an auxiliary electrode, if it isn't an electrode that is not required to be there for the system to be grounded?