Is this Equivocation, Circular Reasoning, or Valid? (Words = "exist" / "existence")

Can you settle a bet for us? Here is the paragraph:

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.

One of us contends this is an axiom, a statement of undeniable truth. The other of us contends this paragraph is a classic case of equivocation on the meanings of the words "exist" and "existence," and that the argument is circular even on its own terms.

What do you think? Thanks for giving this some thought.

Perhaps not germane, but usually "cosmos" is said to be all that exists, accounting for the possibility of more than universe. This would cover the possibility of our universe coming into existence.

So "all that exists" is used interchangeably with "existence"? This seems odd. People talk about their own existence which refers to just one thing that exists... not everything. I, at one point, had no existence. The Mona Lisa, at one point, had no existence.

So if the axiom is said to be that existence always existed... I think this is a bit of a word game and I am not sure what the point would be. Perhaps to show that a god is not needed? The god must "exist" and be part of "existence" so again, not sure where this goes.

Perhaps some others might offer more insight here.

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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.

I have been following this one with interest. I do not see how the paragraph could possibly be accepted as an axiom. It is certainly not a statement that everyone believes is true and accepts as a priori knowledge. Further, based on what we know of the universe is almost certainly not true. Given this, someone arguing this position is committing a fallacy - an error in reasoning. So, the question is what fallacy? From my understanding the arguer would be committing an ambiguity fallacy with a side order of argument by gibberish.

i think in some loosely based phrasing the word existence may in some way refer to "universe". Like if someone says "Unicorns are something most people believe could never be found in all of existence". I think I have heard the word used this way and it seems as if they mean in all the universe. But as a general rule I would not think it can be interchanged with universe frequently because it's just to vague.
I got this from dict.com. No. 4 would suggest that all that exists (exists) would be redundant when existence exists is used. Though I could be reading it incorrectly. 4 also implies it can be interchanged with universe. Kind of weird because it doesnt really seem to be a popular use, but I could be wrong.
Existence Def..
noun
1.
the state or fact of existing; being.
2.
continuance in being or life; life:
a struggle for existence.
3.
mode of existing :
They were working for a better existence.
4.
all that exists :
Existence shows a universal order.
5.
something that exists; entity; being.