Quote by A. Neumaier
in quantum computing they clearly distinguish between measurements and quantum circuits of the kind we discussed, so there must have been a misunderstanding since you wrote:
Quote by Hurkyl
I am under the impression that it's rather standard to allow "measurement" to apply to the indefinite case as well.


Since you apparently quit the discussion just at the point where the crucial gap in your argument had been identified, let me summarize the findings of our extended discussion:
Our CNOT gate discussion started with your claim that it is a measurement device:
Quote by Hurkyl
Let's start with something possibly very simple. I consider a CNOT gate ( wikipedia link) a measuring device. It measures the qubit on its control line, and records the result of measurement by adding it to the target line.

Our discussion revealed that
if each target line is initialized with a definite zero state, CNOT gates can be used to construct an ancilla for a sequence of quasimeasurements, such that the reduced density matrix on the output control line is decohered, i.e., diagonal. Therefore different observers see the same result, conditioned on a particular measurement result for one observer.
But this doesn't hold for CNOT gates that are differently prepared. This shows that the CNOT gate by itself is not a measurement device, but only the dissipative system that consists of
the CNOT gate together with another gate that prepares the target line in a definite zero state. The latter requires already a definite outcome of a measurement, and hence must be itself a measurement device.
Indeed, in quantum information theory, one has specific measurement gates that perform a binary projective measurement and produce a definite outcome. These gates exist as real devices, and are necessary for any quantum information technology.
Thus while CNOT gates explain the working of decoherence in a very elegant and simple way, they  like decoherence itself  do not explain the working of measurement gates (or any other measurement devices).