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.