Expanding two rotation operators

I have the following operators acting on two qubits, denoted as $(1)$ and $(2)$:

$$T_1=displaystyleexpleft(-ifrac{pi}{4}Zotimes Zright)cdot R_z^{(1)}left(frac{pi}{2}right)cdot R_z^{(2)}left(frac{pi}{2}right),quad T_2=R_y^{(2)}left(-frac{pi}{2}right)cdot C_Zcdot R_y^{(2)}left(frac{pi}{2}right).$$

I should prove that up to global phases $T_1=C_Z$ and $T_2=C_X$ (the controlled $Z$ and $X$ gates).

For convenience, I omit the labels for the qubits, which should be clear by the ordering. Using the usual expansion for rotations, I ended up finding

$$T_1=frac{1}{2sqrt 2}(mathbb Iotimesmathbb{I}-iZotimes Z)(mathbb Iotimes mathbb I-iZotimesmathbb I)(mathbb Iotimes mathbb I-imathbb I otimes Z)=frac{1+i}{sqrt 2}begin{pmatrix} -Z & mathbb Iend{pmatrix}, T_2=frac{1}{2}(mathbb Iotimes mathbb I+imathbb Iotimes Y)C_Z(mathbb Iotimes mathbb I-imathbb Iotimes Y)=begin{pmatrix} mathbb I & & Xend{pmatrix}.$$
So, I’ve checked the calculations a few times and it looks like there is something wrong with the first operator. Here’s the catch: there was a mistake in the original $T_2$ I was given that I had to change in order to find the correct sequence to obtain a $C_X$, and there might be a similar problem with $T_1$ (unless I am of course making some mistake somewhere). Does anyone see what the issue may be?

(As a reference, $T_1$ comes from the interaction shift in NMR computing, while $T_2$ from Wineland’s experiment with trapped ions).