If you tried to bake a cake with a recipe and/or knowledge of what ingredients go into a cake and how to put them together, but mis-measured the eggs/used high-protein flour and so ended up with a shitty cake I would cry no-true-Scotsman when someone said you weren't making a cake.
This is quite ironic, considering your sig:
Analogies don't equal equalities, they are merely somewhat analogous.
This is not a useful analogy of the one-time-pad issue, because in that case, the distinction is not predicated on an ad-hoc definition.
The essence of the no-true-Scotsman fallacy is its arbitrary and entirely self-referential circularity: the definition of trueness implicitly adopted by the fallacy's maker is exactly that which (in his mind) makes his argument true - nothing more, nothing less. In particular, it is not derived from any consideration beyond the fallacious argument, and it is not a useful definition in any wider context.
You claim that it is a no-true-Scotsman fallacy to say that an encryption key is not a one-time pad if it is used more than once.
Firstly, it is important to note that this is a narrow claim about semantics: what, exactly, can the the phrase 'one-time pad' be used to denote? It is not a claim about encryption (even though it is made in the context of encryption) because the facts of encryption do not depend on whether this claim is correct.
For your claim to be correct, the assertion 'an encryption key that has been reused is not a one-time pad' must be ad-hoc, introduced solely for the purpose of making an argument look valid, but it is the opposite of that. As you yourself point out, non-reuse is essential to the purpose for what one-time pads are created and used, so the assertion is predicated on a very meaningful distinction. Therefore, it is not a no-true-Scotsman fallacy.
You may think you are still right on the usage issue. Rather than take a position on it, the rest of us can ignore it, because we can make ourselves perfectly clear to one another by considering the context in which the phrase is used. In particular, your original claim that the citation (about the security of one-time-pad-encrypted messages) was incomplete is still wrong, as it very adequately covered the ways in which a key intended to be a one-time pad can be misused.