$\newcommand{\diff}{\mathrm{d}}$The measure $\diff x\, \diff y\, \diff \theta / y^2$ is a product measure. It is the product of $\diff x\, \diff y / y^2$ (area in the hyperbolic plane) and $\diff\theta$ (length in the unit circle in the tangent plane).

For the other measure, we consider the semicircle $L = L_{\alpha, \beta}$ which is perpendicular to the real axis in the complex plane and which has end points at $\alpha$ and $\beta$.  Breaking the symmetry of the situation, we assume that $\alpha < \beta$. Then the centre of $L$ is the point $x = \frac{1}{2}(\alpha + \beta)$.  Also, the highest point of $L$ has $y$-coordinate $y = \frac{1}{2}(- \alpha + \beta)$.  So $\diff x = \frac{1}{2}(\diff\alpha + \diff\beta)$ and $\diff y = \frac{1}{2}(-\diff\alpha + \diff\beta)$.  Thus $\diff x\,\diff y = \frac{1}{4} \diff\alpha\, \diff\beta$.  We deduce that $$\frac {\diff x\, \diff y}{y^2} = \frac{1}{4} \frac{\diff\alpha\, \diff\beta}{((\alpha - \beta)/2)^2} = \frac{\diff\alpha\, \diff\beta}{(\alpha - \beta)^2}.$$ Finally, $t = \theta$ and so $\diff t = \diff \theta$.