I recommend you to read this https://www.iqraonline.net/translations/bidayah-al-ḥikmah-commentary-study-guide/
The groups that believe the meaning of existence is equivocal largely rely on a proof that resorts to causality. They argue that all existences have a causal relationship with each other. For those who ascribe to the view that existence has a different meaning for each existent, they argue that every cause and effect have differing existences. Thus, when a cause brings an effect into existence, their existences differ and as such there are just as many meanings to existence as there are levels in the causal chain of all existence.
As for the view that existence has one meaning for God and another for all other existents, this view has been espoused by the likes of Qāḍī Sa’īd Qumī.7 These sorts of scholars believe that we cannot understand things that are predicated to God. For example, if one says that “God is all-knowledgeable”, we can only really understand that God is not ignorant as we cannot conceive what it means to be all-knowledgeable. They argue that it would be incorrect for us to assume what it means to be all-knowledgeable as the Qur'an has said, [42:11] Nothing is like Him. According to them, the same goes for existence. We cannot conceive what it means for God to exist, all we can know is that He is not non-existent.
Their evidence for this view also refers to causality. They argue that if existence had one meaning then God (a necessary existence) and all other existences (which are possible existences) would have some sort of homogeneity (sinkhīyyah) and such a thing is impossible. That is, nothing can be similar to God and as such, existence must have different meanings.
The answer to this view (which ‘Allāmah has not mentioned in Nihāyah) is that not only is such a homogeneity (sinkhīyyah) possible, rather it is necessary for there to be a similarity. Ever cause and effect must have similarity. In fact, if there were not similarity, then a causal relationship would become impossible (this will be explained later).
If someone claims that the meaning of God existing is different than that of possible existences existing, we will reply by asserting that when we say a thing other than God exists, it means that the thing is an existent.
If existence has a different meaning for God, then we will ask what meaning it has. If it does not mean that God exists or that he is an existent, then it must mean that God does not exist. This is because existence and non-existence are opposites and for something to both not exist and exist at the same time would deny the law of the excluded middle.
As such, claiming that the meaning of existence for God is different than the meaning of existence for anything other than God would result in the belief that God does not exist. However, no proponent of the view that existence is an equivocal term believes that God does not exist.
Furthermore, if someone claims that we cannot understand the meaning of existence that is used for God then this will cause any possible understanding of God to go under question. That is, if we cannot prove God’s own existence than we cannot even discuss his qualities or progress any further in any discussion about him. As such, we would have to put a stop to rationality even though the intellect perceives that existence has one meaning that is the same when predicated to all existences.