In my opinion, therian names are fine as long as you understand that you are your theriotype. I do have a therian name, that name being Evergreen, and I do have a reason for said name:
Basically, I believe that in this life, my soul is part human and part lion. I also like to think that your name is a part of your soul, because it's a part of your identity and who you are. But, I have two parts to my soul. My human side of the soul has a human name, but since I also have a lion side to my soul, when I name that something different it's still MY name. For me personally, it would feel strange for both sides of my soul to have the same name.
There are also spiritual/symbolic reasons for it but that's kind of hard to put into words and it's also pretty personal.
But, I can definitely see where you are coming from, and it could certainly cause confusion for newer therians and make them think that your theriotype is a different entity.
At least in my experience, your therian name is also your name, and only when one does not go by their therian name, it generally may stray from the definition of therianthropy.
(Sorry if this is badly written, I just woke up
)