We study the Finite-Dimensional Distributions (FDDs) of deep neural networks with randomly initialized weights that have finite-order moments. Specifically, we establish Gaussian approximation bounds in the Wasserstein-$1$ norm between the FDDs and their Gaussian limit assuming a Lipschitz activation function and allowing the layer widths to grow to infinity at arbitrary relative rates. In the special case where all widths are proportional to a common scale parameter $n$ and there are $L-1$ hidden layers, we obtain convergence rates of order $n^{-({1}/{6})^{L-1} + ε}$, for any $ε> 0$.