For clarity in this thread, I'm going to start capitalizing terms defined in Article 100 (I may miss some, but those definitions aren't really critical to the discussion).
With "smaller circuits" I'm not attempting to make up a new defined term, I'm just talking about a circuit that is less than the full Branch Circuit.
The crux of our disagreement is whether an incomplete part of a Branch Circuit is itself a Branch Circuit, or whether it is just a circuit. I'm reading the definition of "Branch Circuit" to mean the entire Branch Circuit; if you just take an incomplete part of it, you have a circuit, you have circuit conductors, it is made up of Branch Circuit conductors, but it is not by itself a Branch Circuit.
210.12 refers to the rating of the Branch Circuit. So you have to consider all the Branch Circuit conductors when determining the Voltage of the Branch Circuit. The definition of Voltage (Of A Circuit) makes it clear the MWBC in my example has a Voltage of 240V. That means 210.12 doesn't apply.
The requirement in 406.4(A) you've been referring to reads "grounding-type receptacles shall be installed only on circuits of the voltage class and current for which they are rated," and it does not use the term Branch Circuit. It uses the term circuit, which can be a small portion of a Branch Circuit. So again, the 240V Branch Circuit supplies a 120V circuit that supplies the receptacle. No 406.4(A) violation.
I don't see any other way to reconcile the definitions without reaching a contradiction.
Cheers, Wayne