TITLE: Does the notion of "rotation" depend on a choice of metric?
QUESTION [7 upvotes]: 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).

REPLY [6 votes]: 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.