Proper-motion distance ($d_M=v/\dot{\theta }=(1+z)d_A$)

A source-frame transverse physical speed $v$ produces an observed angular rate $\dot{\theta }$. cosmological time dilation converts source time to observer time, so this operational distance equals $(1+z)$ times the angular diameter distance. In an FLRW geometry it is the transverse comoving distance $R_{0}S(\chi )$. It should be derived with the time convention stated, rather than interpreting the source speed and angular rate as measurements made by the same clock.