There are certian names today, which are cultural names one has. For many english speaking people, they name their children George, or Tony. This does not mean they are naming their kids after George Bush, or Tony Blair, many name their children this, but are firmly against George or Tony.
Only after decades or centuries, - but perhaps a century or more- did the name begin to transition from being simply cultural, to being strongly associated with a personality and only that personality.
In todays age, naming your child Umar or Uthman almost always will be linked to the Caliph, Umar, and Uthman.
However, in those days, we know it was almost certian these were common names, people were named Umar and Uthman before Umar and Uthman, after him, during his life time, unconnected to him etc.
Perhaps you may have had people who named their sons in honour of these two, but in those days, no-one would have immediately jumped to the conclusion they have named their sons after the two caliphs, but rather two quite common names.
In todays society however, due to the prominence these two have gained, the respect, the names Uthman and Umar are almost exclusively linked to the two Caliphs.
As i have shown in my post, shia's everywhere in Muharam condemn the wicked tyrant, Ubaydallah Ibn Ziyad, the butcher of Hussain a.s, and the one who played with his decapitated head with a stick. I have yet to see a shia name their sons Ubaydallah. Yet, the grandson of Hussain a.s and the fith Imam a.s named one of his sons Ubaydallah a.s, and his grandson, the seventh Imam a.s also named one of his sons Ubaydallah.
In those days, it was a common name, and no-one will argue they named them after arguably the most notorious ubaydallah. However today, culture has changed and if you name your son Ubaydallah, it is really linked to the tyrant.
In shia hadith, we find some called 'Yazid', and are followers of the Ahlulbayt asws. Many Umars too, and even Muawiyahs afaik, and many Abu Bakr's. Some are shia's, are followers of the Imams a.s and narrate ahadith from them.