So you're assuming the definitions we would use will cause this problem before they've even been established? Hmm.
Not really. As I´ve said before: even though the term "God" isn´t copyrighted and everyone can use it freely as they wish, there are some common elements to the classical definitions. I am assuming that those "thousand years" of discussion mentioned in the argument dealt with those god concepts (maybe, though, the argument you presented isn´t talking about thousand years of attempts to logically decide the existence of
such a "God" but a completely different "God" of a fringe definition. That´s possible, but that would be important to know for valuating the argument. That´s why I am insisting that it´s important to have a definition.)
Or, IOW: What I have been assuming is that those "thousand years of..." are referring to those very discussions that I - as someone reasonably well versed in the traditional occidental philosophies and theologies - am fairly familiar with. (It´s possible that the argument actually refers to e.g. some age old discussions within an African tribe concerning the existence of some Raingod or something. That would surely make a difference for the appropriate way to approach the argument. However, I find it a bit discomforting when at the same time you refuse clarification but accuse me of making assumptions. It´s not like I would like to make assumptions - but in lack of clarification I am left with going my best guesses).
Many of those core aspects of (classical occidental) god concepts are
1. ex-negativo (there can´t be evidence for God by our standards - emphasis on faith, God is beyond time and space, God is non-physical, God is mysterious, unintelligible by the human mind etc. etc.), and at the same time
2. are postulating the subject to be "supernatural".
(in which 1 and 2 can´t seem to always be kept entirely apart)
These definitions are picturing an in many ways exceptional entity that excludes its existence from being decided by the means to our disposal. To pick an extreme, for clarification (not saying that this is necessarily part of those classical god definitions): Once I have defined a subject to be "beyond logic" it should be obvious that the definition itself renders a logical approach obsolete.
Of course, given the fact that everyone can use "God" as they see fit (and in case the definition used would be irrelevant for the discussion of the argument presented), I don´t quite understand why there were thousand years of failed attempts to establish "God´s" existence.
I could do it in a heartbeat.
I could define "God" as this pencil lieing there in front of me, and I would have all evidence for "God´s" existence on my side.
Or, to pick a more realistic example, I could define "God" as the "first cause", and even if the universe were uncreated, popped into existence out of nothing, had no cause, created itself, has existed forever (or whatever other scenario without a supernatural entity can be fathomed), I could still conclude that the universe is the first cause = "God". That´s a very simple way of defining a "God" into existence That is logically necessary.
OTOH, certain god concepts can easily be decided to point to a non-existent entity (and have been so during these thousand years of discussion) by logic alone: they are self-contradictory.
Thus, premise #1 is questionable in many ways.
Bottom lines:
1. If the definition doesn´t matter, there isn´t any problem to decide that God exists.
2. If, as is a core premise of the argument as presented, there have been thousand years of exhaustive but unsuccessful discussion of the problem, we must assume that they must have been based on (a) somehow defined concept(s) - or else these discussions couldn´t even have taken place. The nature of the discussions and their result (or lack of a conclusive result, in this case) necessarily have to do with the nature of this definition (these definitions). Therefore alluding to the fact that there have been exhaustive discussions but at the same time refusing to take a closer look on the definitions that shaped the discussions and therefore the results would render the argument inconsistent.
Anyway, the problem with #3 (introducing the "unknowability" aspect of God in order to establish that the inability to decide God´s existence is somewhat evidence for this unknowable God) is perfectly illustrated by this old joke:
Two men meet.
A: What are you chewing?
B: Apple pips.
A: Why the heck are you chewing them - they are disgusting.
B: Yes, but they keep the lions away.
A: But...but there aren´t any lions.
B: See?
A good point. I suppose I would need an example where logic played a role in determining existence before the argument would have any traction. I don't think I have any way around that one. IMO it's this that finally makes the argument crumble, not the definition thing you started with. So, thanks for putting that aside and pointing this out.
I think the two are intertwined. When/if "God" is defined as having no evidence by our standards (and, as far as I can see, this is a common definition trait of all "supernatural" concepts), this definition makes it logically impossible to even try to approach the question with logic because according to the definition there can be no evidence to deal with logically.
That begs a new question, though. Do definitions do anything except classify what we already assume to exist?
Not quite sure what this has to do with anything, but it´s an interesting question nonetheless. I have never pondered it.
My first thought would be:
Yes, sure we can define a word to signify something that we don´t assume to exist. E.g. I could define "prufatosk" as a "pink elephant with wheels" without assuming that such an entity exists.
However, I don´t see much need for anyone to do that - except maybe for artistic purposes. Or for mind games. Or for purposes of fraud ("I can offer you a house in Paris right on the beach of the Pacific".)

Or, IOW: Yes, our classifications can - knowingly,intentionally, purposefully - contain empty classes.