This information I got from Oxford Grammar by M. Swan, tells me that the difference between identifying and non-identifying clauses is just whether the topic that the speakers are talking about is conveyed or not.
For instance,
The book that got the prize for best first novel was written by my landlady.
The sentence above contains an identifying relative clause.
'Wild cherries', which got the prize for best first novel, was written by my landlady.
Whereas, the second sentence above contains a non-identifying relative clause.
For more examples, here's the screenshot of the e-book I took :
I notice that the non-identifying relative clauses are preceded by names. Related to my question, if the preceded subject is a name, is it always called a non-identifying clause?
P.s. I've been trying to find a counterexample, i.e. if it's not a name, but I haven't come up with something.