Holonomy representation: is it actually a class of representations? In D. Joyce's book "Riemannian Holonomy Groups and Calibrated Geometry" (2007) the author writes that if $\nabla$ is a connection on a vector bundle $E$ (over a connected base) with the fibre $\mathbb R^k$ then the holonomy group $\mathrm{Hol}(\nabla)$ comes equipped with a natural representation on $\mathbb R^k$, or equivalently, with an embedding of $\mathrm{Hol}(\nabla)$ in $\mathrm{GL}(k,\mathbb R)$. He calls this representation the holonomy representation.
But actually $\mathrm{Hol}(\nabla)$ is defined as a subgroup of $\mathrm{GL}(k,\mathbb R)$ up to a conjugation.  Does it mean that actually the holonomy group comes equipped with a family of representations on $\mathbb R^k$ and "the holonomy representation" is actually a class of representations?
On the other hand, it's clear that the holonomy group $\mathrm{Hol}_x(\nabla)$ with fixed basepoint comes equipped with a natural representation on $E_x$, the fibre of $E$ over $x$.

Yes. The holonomy representation is an isomorphism class of representations. 
