Is projective space really a moduli space for lines through the origin?

Question

The Wikipedia page for Moduli spaces states that real projective space $\mathbb{RP}^n$ is a moduli space which parametrizes the space of lines in $\mathbb R^{n+1}$ passing through the origin. However, any line $L$ passing through the origin in $\mathbb R^{n+1}$ has non-trivial automorphisms given by stretching the line (i.e., those linear transformations which have $L$ as an eigenvector). It's well known that the existence of non-trivial automorphisms precludes the existence of a moduli space. What is going on here?