Two possible candidates for the moduli space of inner products on $\mathbb{R}^n$ come to mind:
the space $S_n$ of positive-definite symmetric real $n\times n$ matrices; or,
the classifying space $BO(n)$.
Why 1: the points of $S_n$ are in clear bijection with inner products on $\mathbb{R}^n$.
Why 2: due to the groupoid equivalence $[X,BO(n)] \simeq (\text{rank-$n$ real v.b.'s on $X$ with bundle metric})$, applying $X = \mathrm{pt}$, we see that the groupoid of rank-$n$ real v.b.'s equipped with bundle metric over $\mathrm{pt}$ (which should correspond to inner products on a vector space isomorphic to $\mathbb{R}^n$) is equivalent to the fundamental groupoid of $BO(n)$; in particular inner products on a real dimension-$n$ vector space correspond to points of $BO(n)$.
Is $BO(n)$ "larger" because it also carries information about inner products on all possible dimension-$n$ vector spaces, instead of just $\mathbb{R}^n$?
What would be the appropriate formal context in which to distinguish these two "types" of moduli spaces?
Apologies if my reasoning is informal and/or mistaken; I have little familiarity with the concept of moduli space.
The space of positive-definite symmetric real $n \times n$ matrices is the moduli space of inner products on $\mathbb{R}^n$, in a very concrete sense. $BO(n)$ is, in a less concrete (homotopical) sense, the moduli space of inner products on $n$-dimensional real vector spaces; note the difference here (we haven't chosen a basis, or equivalently, we haven't chosen an isomorphism of such a vector space with $\mathbb{R}^n$).
This is in fact what you've written, so just take the time to notice that these are not the same thing.