According to Hartshorne, Foundations of Algebraic Geometry, $PGL(2, \mathbb R)$ denotes the group of automorphisms of $\mathbb{RP}^2$, the real projective plane, and is therefore a quotient of $GL(3, \mathbb R)$. That is, there appears to be an off-by-one convention in the use of the numerical index; in this sense the convention is similar to the fact that $\mathbb{RP}^n$ is a quotient of $\mathbb R^{n+1} \setminus \{0\}$.
Hartshorne writes explicitly:
Hence an element of $PGL(2, \mathbb R)$ is represented by a $3 \times 3$ matrix $A = (a_{ij})$ of real numbers, with $\det A \ne 0$, and two matrices $A, A'$ represent the same element of the group if and only if there is a real number $\lambda \ne 0$ such that $A' = \lambda A$.
But the general consensus (as evidenced by the Wikipedia article and by the comments on another question) seems to be that $PGL(n, k)$ is a quotient of $GL(n, k)$, not a quotient of $GL(n+1, k)$. That is, the index $n$ in $PGL(n, k)$ designates the index of the parent space of which it is a quotient, not the dimension of the projective space on which it acts.
Are both conventions widely used, but in different contexts? (For example, perhaps algebraic geometers use one convention, and group theorists use the other?) Is one perhaps more archaic? (I note that Hartshorne's book is 54 years old; perhaps it reflects an older convention?) Is there a consensus, nowadays, on which of these is "right"?
Or is it possible that I am misunderstanding something, and that in fact there is no disagreement here at all?
Edited to add: I just checked Hartshorne's Algebraic Geometry to see if he uses the same convention in that other, better-known book, and can confirm that he does:
p. 151:
Example 7.1.1 (Automorphisms of $\mathbf{P}^n_k$). If $\|a_{ij}\|$ is an invertible $(n+1)\times(n+1)$ matrix of elements of a field $k$, then $x_i' = \sum a_{ij}x_j$ determines an automorphism of the polynomial ring $k[x_0, \dots, x_n]$ and hence also an automorphism of $\mathbf{P}^n_k$. If $\lambda \in k$ is a nonzero element, then $\| \lambda a_{ij} \|$ determines the same automorphism of $\mathbf{P}^n_k$. So we are led to consider the group $PGL(n, k) = GL(n+1, k)/k^*$, which acts as a group of automorphisms of $\mathbf{P}^n_k$.
So at least Hartshorne is self-consistent in this regard.