Okay. There are some claims which are logically inconsistent, but that depends on the assumptions you make, do you agree?
In mathematics, it is possible to construct different logically consistent sets of claims based on different assumptions. Then whether a statement belongs to the "true" set of claims would depend on which assumptions you made. The classic example is non-euclidean geometry. Again, I don't know what you really understand and don't understand, so forgive me if I am telling you something you already know.
The reason we argue over definitions is because these are our assumptions. It is always possible to claim a given statement is "true" (logically consistent) if we choose our assumptions or definitions correctly. Most people aren't going to argue over the definition of "car", but as you saw in the last thread we can argue endlessly over the definition of "atheist".
Similarly, I can posit a definition of "deity" by which there are supernatural entities which are not deities. I suspect that Buddhists use this definition.
Yes, I understand.
What can be logically inconsistent about a definition?
A lot. If the definition includes premises that contradict other premises (within the same definition) or were conclusions don't follow from these premises, we create inconsistencies.
In the atheist definition posed, all was well and dandy until a logical inconistency appeared.
It is logically inconsistent to state no beliefs in gods, and affirm no mental process is required to state that.