Consider the statement:
The Euclidean metric on $\mathbb{R}^n$ is rotationally invariant.
I interpret this to mean (is this interpretation correct?):
The Euclidean metric on $\mathbb{R}^n$ is invariant under the action of the orthogonal group $O(n)$.
However, the orthogonal group $O(n)$ is defined in terms of the Euclidean metric (as the group of all self-maps $\mathbb{R}^n \to \mathbb{R}^n$ which preserve Euclidean distance and fix the origin).
This suggests that we are implicitly using the following definition of "rotation":
Rotations are the set of all (orientation-preserving) isometries of $\mathbb{R}^n$ which fix the origin.
Question: Why is the first claim "the Euclidean metric on $\mathbb{R}^n$ is rotationally invariant" noteworthy/not trivial if we are implicitly using this definition/notion of rotation?
(I.e., of course the metric is preserved by a group of isometries.)
When we define "rotations", how are we not implicitly choosing a preferred metric on $\mathbb{R}^n$?
/Question
Clarifying example: In contrast,
The taxicab metric on $\mathbb{R}^n$ is not rotationally invariant.
In other words,
The taxicab metric on $\mathbb{R}^n$ is not invariant under the action of $O(n)$.
But what if we consider, instead of $O(n)$, what I will call $T(n)$ ("taxicab orthogonal group") of all self-maps $\mathbb{R}^n \to \mathbb{R}^n$ which preserve taxicab distance and fix the origin?
It seems fairly clear that we have:
The taxicab metric on $\mathbb{R}^n$ is invariant under $T(n)$.
or in other words
The taxicab metric on $\mathbb{R}^n$ is "taxicab-rotationally invariant".
Note: This is a very dumb question, so if you have any suggestions for how it could be improved, or if it should just be deleted, please say so (nicely).
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.