I wanted to add something quickly.
Classical logic does indeed treat ∃x(x = x) as a logical truth. Likewise, so is ∃x(x = c). However, if ‘c’ is some sort of truncated definite description, as Russell believed, so it would be read as "the x such that...", then it is not a truth of logic, because it might fail to denote. Thus, that symbolization might not always be a truth of logic, and indeed you will see many philosophers use or propose it to signify singular existentials, Quine was no different:
“But for those who do not share his {Russell’s] view on proper names, the restriction [the restriction of applying existence as a predicate to just generalized existentials] could be overcome by adopting Quine's proposal that ‘Socrates exists’ be reparsed as ‘(∃x)(x = Socrates)’.” (See the Miller article)
Of course I neither meant to use ‘c’ as a definite description nor do I follow Quine. I use it simply to refer to Craig. The point is just that someone could use it in such a way without it being a truth of logic, contra Vaz.
I find it useful since it does the work that I need it to do in that instance without committing me to definite descriptions. It is my rejection of definite descriptions that also forces me to reject Vaz’s proposal to treat a singular existential statement such as ‘Craig is’ as this (Ex)Cx. If Vaz wants to treat these statements as philosophically neutral, then he will meet resistance, since the very project of classical logic in philosophy is to regiment or even replace natural language. Thus, they say, it cannot be ignored.
That said, my rejection is not a sincere and dedicated rejection, since I oscillate between the two without much thought when I do philosophy. I note the difference only when they matter to the subject at hand. Unlike Frege and Leibniz, or Russell, or Quine, I see classical logic as a failure in its project to capture the logic of natural language. It is a useful tool sometimes, and that's all the stock I put in it.
If there is some profound ignorance in this, ok. But I can cite two articles in the Stanford that use the same formalization. I can also cite early Frege and even Quine. I'm in good company.