Does the notion of "rotation" depend on a choice of metric?

Most mathematical concepts can be defined in many different ways. In (most) books, great care is taken to ensure there are no clashes among definitions, resulting in a streamlined and concise presentation. But yes, if you put a bunch of definitions and statements in the same bag, you will end up with circularities/trivialities.

Now, one definition of the rotation group $SO_n$ which I like is: $$ \{A \in \mathbb{R}^{n\times n} \,|\, A^T A = I_n,\, \det A = 1\}. $$ It is easy to see that the euclidean metric is invariant under the action of $SO_n$.

So the statement 'the euclidean metric is rotationally invariant' is, in this case, a gentle reminder of one geometric property of $SO_n$.

Note. As noted by Michael, there is no circularity in the examples you provide, only redundancy and triviality.