There is such an interpretation, with a few caveats. Essentially, there is a canonical connection on a certain vector bundle for which the "principle part" of the curvature is the Weyl tensor in dimensions $n\geq4$, and the Cotton tensor when $n=3$. I will describe this from the point of view of the tractor calculus, but avoid introducing unnecessary bundles where needed. This can also be described using the Fefferman--Graham ambient metric or using Cartan connections. This summary mostly follows Bailey--Eastwood--Gover, though Armstong and articles written by Gover are also good references. I use abstract index notation throughout.
First, we define conformal densities. Given a conformal manifold $(M,c)$, a conformal density of weight $w\in\mathbb{R}$ is an equivalence class of pairs $(g,f)\in c\times C^\infty(M,c)$ with respect to the equivalence relation $(g,f)\sim(e^{2\Upsilon}g,e^{w\Upsilon}f)$. Let $\mathcal{E}[w]$ denote the space of conformal densities of weight $w$. We similarly define $\mathcal{E}^i[w]$ as the space of equivalence classes of pairs $(g,v^i)\in c\times\mathfrak{X}(M)$ with respect to the equivalence relation $(g,v^i)\sim(e^{2\Upsilon}g,e^{w\Upsilon}v^i)$. Here $\mathfrak{X}(M)$ is the space of vector fields on $M$.
Next, we define the space of sections of the standard tractor bundle. Fix a metric $g\in c$. Define $\mathcal{T}_g^A=\mathcal{E}[1]\oplus\mathcal{E}^i[-1]\oplus\mathcal{E}[-1]$. Given another metric $\hat g := e^{2\Upsilon}g\in c$, we identify $(\sigma,v^i,\rho)\in\mathcal{T}_g^A$ with $(\hat\sigma,\hat v^i,\hat\rho)\in\mathcal{T}_{\hat g}^A$ if $$ \begin{pmatrix} \hat\sigma \\ \hat v^i \\ \hat\rho \end{pmatrix} = \begin{pmatrix} \sigma \\ v^i + \sigma\Upsilon^i \\ \rho - \Upsilon_j v^j - \frac{1}{2}\Upsilon^2\sigma \end{pmatrix} . $$ (Recall these are densities, so exponential factors are suppressed.) The space of sections $\mathcal{T}^A$ is the result after making this identification. Note that the top-most nonvanishing component is actually conformally invariant modulo multiplication by an exponential factor. Because of this, we call the top-most nonvanishing component the projecting part.
There is a canonical connection on (the vector bundle whose space of sections is) $\mathcal{T}^A$, the standard tractor connection, which, given a choice of metric $g\in c$, is given by the formula $$ \nabla_j \begin{pmatrix} \sigma \\ v^i \\ \rho \end{pmatrix} = \begin{pmatrix} \nabla_j\sigma - v_j \\ \nabla_j v^i + \sigma P_j^i + \delta_j^i\rho \\ \nabla_j\rho - P_{ji}v^i \end{pmatrix} . $$ Here $P_{ij}=\frac{1}{n-2}\left( R_{ij} - \frac{R}{2(n-1)}g\right)$ is the Schouten tensor and $n=\dim M$. It is straightforward to check that this is well-defined, in the sense that it is independent of the choice of matrix $g\in c$.
Given a metric $g\in c$, it is straightforward to compute that $$ (\nabla_i\nabla_j - \nabla_j\nabla_i)\begin{pmatrix} \sigma \\ v^k \\ \rho \end{pmatrix} = \begin{pmatrix} 0 & 0 & 0 \\ C_{ij}{}^k & W_{ij}{}^k{}_l & 0 \\ 0 & -C_{ijl} & 0 \end{pmatrix} \begin{pmatrix} \sigma \\ v^l \\ \rho \end{pmatrix} . $$ This is conformally invariant by construction. The "3-by-3" matrix is the tractor curvature, and its projecting part is $W_{ij}{}^k{}_l$ when $n\geq4$ and $C_{ij}{}^k$ when $n=3$. Standard interpretations of holonomy then give the interpretation of the Weyl tensor in terms of parallel transport around infinitesimal loops that I indicated in the first paragraph.
Finally, given your bullet points, let me emphasize that the signature of $c$ plays no role here, and everything is manifestly conformally invariant.
Added in response to a comment. There are many geometric motivations for introducing the standard tractor bundle. One is that the conformal group of the sphere is $SO(n+1,1)$, so it makes sense that the right replacement of the tangent bundle of a conformal $n$-manifold should be a bundle of rank $n+2$, as is the standard tractor bundle. Note that the metric on $\mathcal{T}$ has signature $(n+1,1)$, assuming we start with a conformal manifold of Riemannian signature (if $c$ has signature $(p,q)$, the metric on the standard tractor bundle has signature $(p+1,q+1)$).
Another motivation comes from the ambient metric. First, note that the flat conformal sphere $(S^n,c)$ (i.e. the conformal class of the round $n$-sphere) can be identified with the positive null cone $\mathcal{N}$ centered at the origin in $\mathbb{R}^{n+1,1}$. This is done by noting that the projectivization of $\mathcal{N}$ is $S^n$ and identifying sections of $\pi\colon\mathcal{N}\to S^n$ with metrics in the conformal class $c$ by pullback of the Minkowski metric. (Incidentally, this leads to a proof that $SO(n+1,1)$ is the conformal group of $S^n$.) In this case, a fiber $\mathcal{T}_x$ of the standard tractor bundle is identified with $T_p\mathbb{R}^{n+1,1}$ for some $p\in\pi^{-1}(x)$; this is made independent of the choice of $p\in\pi^{-1}(x)$ by identifying tangent spaces at points subject to a homogeneity condition matching that of $\mathcal{E}^i[-1]$ above. The standard tractor connection is then induced by the Levi-Civita connection in Minkowski space, after making some identifications.
For a general conformal manifold $(M^n,c)$ of Riemannian signature, Fefferman and Graham showed that there is a "unique" Lorentzian manifold $(\widetilde{\mathcal{G}},\widetilde{g})$ which is "formally Ricci flat" and in which $(M^n,c)$ isometrically embeds as a null cone. Here formally Ricci flat means that the Ricci tensor of $\widetilde{g}$ vanishes to some order, depending on the parity of $n$, along the null cone, and I write unique in quotes because the metric is only determined as a power series to some order along the cone, and this up to diffeomorphism. One recovers the standard tractor bundle and its canonical connection from that of $(\widetilde{\mathcal{G}},\widetilde{g})$ as in the previous paragraph. See Fefferman--Graham for details, or Čap--Gover for a detailed description of the relation between the tractor calculus and the ambient metric, including the identifications I didn't detail. A similar construction for other signatures works, consistent with what is described in the previous paragraph.