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Existence exists. When the universe is defined as all that exists, and used interchangeably with existence, there is no cause or explanation for the origin of existence. It exists and has existed eternally. It did not come into existence. One can play word games, but in the end one word must encompass the concept of “all that exists.” Pick whatever word you want, but the word “existence” is commonly used for this, but many also use the term “universe” interchangeably with existence. Because of that, I prefer the word existence to mean all that exists.
The first (and perhaps the biggest) problem lies in the fact that "exists" and "existence" (the former a linguistic predicate, the latter a noun, normally indicating a property of some sort) haven't been defined. Thus "existence exists" might well involve a logical problem, but it's not clear which. What follows, therefore, would be disputed by someone who had a different understanding of the terms.
I take "existence" to be a second-order property, and "exists" to be a second-order predicate; that is, to say that
x exists, or has existence, is to say that the properties of
x are co-instantiated in the world (one might, therefore, say that "exists" and "existence" tell us about the world rather than about the thing referred to as "
x ").*
On this account (and, to be honest, on most others) it makes no sense to say that existence exists. Nor, though, does it make sense to say that the world, or universe, exists, as that would mean that the properties of the world are co-instantiated in the world. Rather, individual things or states of affairs can be said to exist, and the world is the sum total of those individuals.
The argument under discussion is flawed on a variety of grounds, therefore. First, it makes no sense to say that existence exists. Secondly, it's simply untrue that 'many also use the term “universe” interchangeably with existence'. Thirdly, even if it made sense to say that existence exists, that would rule out as senseless the claim that "existence" means "all that exists", as you've already said that existence is just one of the things that exists.
To be honest, it's hard to make sense of the way that the argument is supposed to work, as it involves so many circles, contradictions, and vaguenesses.
* In philosophy "world" is usually used to mean "universe", as in possible-world theory; strictly speaking, the universe is everything there is, but scientists started misusing the term to mean a specific sort of bit of the universe, so that other terms -- such as "multiverse" -- had to be coined.