The name tag given to them is not important. The decisive factor of allegiance depends on their command, before PMU arrival the situation was that the YBS (Yazidi/Ezidi units) were strongly influenced by the PKK given the absence of the PMU/ISF. With the arrival of the PMU they forced these Yazidi units to change command/allegiance.
Yazidis have been at odds with Barzani for many reasons thus they sided with the YPG/PKK, that changed now.
As this article states,
The Kurdistan Workers Party (PKK) in the Yezidi region of Sinjar (Shingal) in northern Iraq, and its local affiliate the Shingal Resistance Units (YBŞ) oppose the presence of the Shia paramilitary Popular Mobilization Units (PMU) that in May captured the south of Sinjar from ISIS.
“I think that the PKK resents the presence of the PMU because the latter represents a threat to the PKK’s long-term influence in Sinjar. Young men have left the YBŞ, just as they are leaving the Peshmerga, to join the PMU,”
http://aranews.net/2017/06/pkk-resents-shia-paramilitary-presence-in-yezidi-region-of-sinjar/
Simply said, before PMU arrival the PKK/YPG was the strongest nearest force for Yazidis(YBS) to ally with, PMU came and took over as the force with the upper hand. PKK/YPG can't do much against PMU thus they lost their influence in Sinjar region.
Maybe you should get rid of Barzani and have PMU rule Qandil as well.
