I would submit the broad, provocative claim that no logical proof of existence can be convincing, because "proof" assumes logic, but logic does not in and of itself talk about what actually obtains in the world. In this type of proof that claims to establish "existence", the proof can be flawlessly correct from a logical point of view, but the premises will invariably be flawed. Typically, it seems the logical form of the argument is presented as the mechanism that will convince, but relying on a logical deduction is not a guarantee in and of itself that something exists or does not exist. The logical form of the argument does not strengthen it because the rules of logic have no bearing on what actually exists and what does not exist.
So, even if an argument is clad in logic, the mere fact that it uses logic is not enough to lend it authority as a claim about anything existing. Existence has to be observed to be established (note: "observation" is replete with difficulties, and there is a serious rabbit hole here about our senses and what really is in the world, but I will blissfully ignore it in this discussion as it does not really change the point I want to make). Logic can very well take premises that are untrue of the world and carry out flawless logical reasoning to conclusions which are ... equally untrue of the world ("all dogs are blue, Rex is a dog, thus Rex is blue" is perfect logic).
In the argument above, as in other claimed "existence proofs", the premises can all be attacked. The first one is an assertion where "limited" and "cause" beg to be analyzed and clarified. The second is an assertion that has no merit as a claim about the world unless you can adduce enough evidence to support it. For the second argument, "god" begs to be defined in the second premise, and the fact that "god is unlimited" is an arbitrary claim, or a circular definition ("god is the one who is by definition unlimited", maybe, but that presupposes that "god" exists, so we can assert claims about properties of "god"? How can you say that god has such and such attributes without e.g. establishing first that god exists?) etc etc.
But more importantly, it is kind of a category mistake to present a logical proof for the existence of anything, because existence is not something you prove, it is something you observe of something in the world. Even if a logical deduction were based on observed premises, you could conclude to a hypothesis, but you could not assert the complete truth of the existence of something. You would have to add a confirmatory step in the form of an observation to be certain that something "exists". The reason for requiring this, is that otherwise we can very well claim that things "exist" that in fact do not "exist", without any end.
Of course, you can claim direct evidence for the existence of god, for example by saying that to you, fine tuning for life is evidence for the existence of god. But you don't need logic to lend extra strength to that claim. Cladding the argument in the form of a logical argument will not in and of itself make it more "true". Maybe it will look more impressive, but it is just a kind of trick to impress the audience, the logic form does not make the facts about the world more or less true.
(post scriptum: I would be surprised if many believers required logic to have faith. Belief in god is probably more immediate and direct than any "logic", and no amount of logic one way or another should interfere with faith, so those "proofs" for the existence of god seem unneeded to me. I am not sure what the usefulness of those "proofs of the existence of god" based on logic really is: they are not needed for the faithful who doesn't need any logic to believe in god, and they probably won't convince the unfaithful by themselves, while for the philosophers, the premises will invariably entail some questionable contingent commitment or other)
(post scriptum 2: another problem here, is that this argument may apply to only some understandings of the idea of "god". There could very well be an understanding of "god" as an entity that is limited in some sense for example. Or god may just be the same as all of nature for some others. So, the argument, if successful, would only establish the existence of one type of god, while other types might also "exist")