Jacobian of hidden state update in backpropagation through time

I am trying to understand the gradients of backpropagation through time for a simple recurrent neural network. In particular this one: https://arxiv.org/abs/1211.5063 (Section 1.1)

(Also here: https://cs224d.stanford.edu/lectures/CS224d-Lecture8.pdf and other blog posts that just refer to the chain rule.

EDIT: https://www.deeplearningbook.org/contents/rnn.html page 379)

The update for the "hidden state" is: $$\mathbf{x}_t=\mathbf{W}\sigma(\mathbf{x}_{t-1}) + \mathbf{W}_{\mathrm{in}}\mathbf{u}_t + \mathbf{b}$$

Calculating the gradients makes sense up to the step when it comes to $$\frac{\partial \mathbf{x}_i}{\partial\mathbf{x}_{i-1}} = \mathbf{W}^\top diag(\sigma'(\mathbf{x}_{i-1})$$

Why is it $\mathbf{W}^\top$? I tried to come to the same result for a simple 2x2 example but I get a different result:

$$\mathbf{y} = \mathbf{W}\sigma(\mathbf{x})=\begin{bmatrix} w_{1,1} & w_{1,2} \\ w_{2,1} & w_{2,2}\end{bmatrix}\begin{bmatrix} \sigma(x_1) \\ \sigma(x_2)\end{bmatrix} = \begin{bmatrix} w_{1,1}\sigma(x_1) + w_{1,2}\sigma(x_2)\\ w_{2,1}\sigma(x_1) + w_{2,2}\sigma(x_2)\end{bmatrix}=\begin{bmatrix} y_1 \\ y_2\end{bmatrix}$$

$$\frac{\partial \mathbf{y}}{\partial \mathbf{x}} = \begin{bmatrix} \frac{\partial y_1}{\partial x_1} & \frac{\partial y_1}{\partial x_2} \\ \frac{\partial y_2}{\partial x_1} & \frac{\partial y_2}{\partial x_2}\end{bmatrix} = \begin{bmatrix} w_{1,1}\sigma'(x_1) & w_{1,2}\sigma'(x_2)\\ w_{2,1}\sigma'(x_1) & w_{2,2}\sigma'(x_2)\end{bmatrix} = \begin{bmatrix} w_{1,1} & w_{1,2} \\ w_{2,1} & w_{2,2}\end{bmatrix}\begin{bmatrix} \sigma'(x_1) & 0 \\ 0 & \sigma'(x_2)\end{bmatrix}=\mathbf{W} diag(\sigma'(\mathbf{x}))$$ This is also what I would expect applying the chain rule (with $\mathbf{z}=\sigma(\mathbf{x})$): $$\frac{\partial \mathbf{y}}{\partial \mathbf{x}} = \frac{\partial \mathbf{y}}{\partial \mathbf{z}} \frac{\partial \mathbf{z}}{\partial \mathbf{x}} = \mathbf{W}\frac{\partial \mathbf{z}}{\partial \mathbf{x}} =\mathbf{W} diag(\sigma'(\mathbf{x})) $$

What am I missing / doing wrong here?

Thank you for your help!