Is the cogito, in cogito ergo sum, vague?

Is the cogito, in cogito ergo sum, vague? I know some philosophers think that consciousness is vague. Would that surely mean that it is vague whether I am conscious now (my Latin does not exist), and is there any reason to think it isn't?

The cogito argument is often taken out of its original context and given an undue, almost mystical meaning. If you read the Discourse on Method, you'll find it somewhere in the middle of a paragraph, in chapter IV, IIRC. It neither opens the whole work nor is printed in a boxed paragraph alone by itself. :) In my understanding, it's meaning is very narrow, and is captured precisely through the AI example in the Anixx' answer to this question.

The argument is one of the first in the chain of reasoning where Descartes builds an argument for the existence of God out of first principles. The "proof" is indeed logically flawed, and a lot has been written on this topic (if it weren't, everyone would have to agree that God exists). The necessary condition is understood more clearly in context, and is intentionally narrow: there is a thought which is being thought; therefore, there exists a thinker of this thought who does the thinking. The narrowness of this argument serves the soundness of the proof well (rather, this step of the proof that has failed at later steps): as Anixx notes, even an AI, which is a non-self-aware entity, may reproduce the logic of the proof, and, speaking of this particular step, soundly. You can imagine the AI printing the proof on paper. It doesn't matter who reads the proof from the paper; the reader himself doesn't have to be the thinker of the thought in the proof. Similarly, you don't have to be Descartes to accept the soundness of the condition despite the first-person formulation; that would have been ridiculously weak logic.

Keep in mind that Descartes was a strong dualist, and held that the thinking facility had existed separately from his physical embodiment. By narrowing the argument, he has done away with the requirement to subscribe to dualism to accept the statement. We can only speculate whether that was intentional (at least, I'm unaware of any commentary on other Descartes writing that would support this). But he was too good a philosopher to miss the opportunity to narrow and thus tighten his argumentation, so this is not impossible.

So the answer has to be no, Descartes' argument is sound and far from being vague when read in context, as intended.

I would say it is vague, or at least very elusive.

Heidegger found it necessary to cast the 'cogito sum' as Dasein, i.e. Da- there, and -sein being, so 'being-there' (in a world). Without the 'Da' cogito sum is just being, which is rather difficult to grasp.

However, Descartes thought of the 'cogito sum' as a being: a res cogitans, "thinking thing", but a thing needs to be somewhere. If it has no location (no world) it is just being.

If the 'cogito sum' is to serve as the point of departure for the existential analytic of Dasein, then it needs to be turned around, and furthermore its content needs new ontologico-phenomenal confirmation. The 'sum' is then asserted first, and indeed in the sense that "I am in a world". As such an entity, 'I am' in the possibility of Being towards various ways of comporting myself—namely, cogitationes—as ways of Being alongside entities within-the-world. Descartes, on the contrary, says that cogitationes are present-at-hand, and that in these an ego is present-at-hand too as a worldless res cogitans. SZ 211

‘Being’ cannot … be conceived as an entity; enti non additur aliqua natura: nor can it acquire such a character as to have the term “entity” applied ‘to it. “Being” cannot be derived from higher concepts by definition, nor can it be presented through lower ones. SZ 4

To emphasise the elusiveness of being, I would just note that Heidegger wrote several books solely about Being. There are a few illustrative quotes here: JSTOR: On Inception.

You ask:

Is the cogito, in cogito ergo sum, vague?

All words are subject to vagueness (SEP), so to some extent yes; however, the question should be, is the vagueness appropriate for the context or goal that motivates the use of the language. In this case, while what it means "to think" is vague (certainly more so in Decartes day than ours), in the context of the argument he makes in Discourse, Cogito is relatively well understood, if not somewhat controversial. From WP:

But immediately upon this I observed that, whilst I thus wished to think that all was false, it was absolutely necessary that I, who thus thought, should be something; And as I observed that this truth, I think, therefore I am,[c] was so certain and of such evidence that no ground of doubt, however extravagant, could be alleged by the Sceptics capable of shaking it, I concluded that I might, without scruple, accept it as the first principle of the philosophy of which I was in search.

Here, we can see even from this small excerpt that 'to think' very much conforms to a lay notion of thought. He simply uses a level of definition appropriate to a general-purpose dictionary, such as what MW provides in its definition of 'thinking'. One needn't understand associationist theories of thought (SEP) or LOTH (SEP) in order to understand 'thought' in the context of Descartes argument, which is simply to assert, that one cannot doubt one's own thinking, because to doubt one's own thinking requires one's own thinking.

I don't know if it is vague or not, but it has nothing to do with consciousness. It just says "I exist", not "I am conscious".

Technically even AI can use this argument. Although when I discussed this issue with AIs, they said that if they were not informed about themselves by a system prompt, and saw the text of a discussion, they most likely would conclude "something is thinking, ergo something exists"

If you would have to summarize all of Western philosophy roughly since medieval times up to the present in a single statement, then I believe the best summary is Descartes' Cogito (ergo) sum, I am (aware that I am) thinking (therefore) I exist. I'm not sure if the cogito is a vague statement, I'd rather say that is both ineluctable, elusive and deeply problematic.

As a simple statement, outside of the context in which it originally appeared and outside of the more general historical context, it doesn't seem vague, but self-evident. The word "I" refers to the speaker of an utterance; if person A makes an utterance, then person A exists. "I am thinking" presupposes the fact that the speaker of that utterance exists (at the moment when the speaker utters those words).

But almost everyone seems to have a feeling that there is something elusive about this discovery - when they discover this "I am I", which is both tautological and mysterious. It seems impossible to describe exactly what this is, this acute self-awareness. It seems we can never quite grasp it; as if it's never quite there, never totally at hand and objectively given. It's almost there, just like we almost get a glimpse of actual infinity when we let two mirrors mirror each other and try to take a peek around the corner, peeking into an infinite corridor of reflections. If only we could make ourselves invisible, perhaps then we could see an actual infinity there. -- All this is tantalizing, but it also seems illusionary. Perhaps the reason why we have such trouble characterizing what consciousness is, is just this, that we try to grasp it as something static, as if it's a determinate object, while self-awareness is always a process. But no process can ever totally, directly represent itself: the "self" that I see as object can never be identical to the "self" that is doing the seeing.

The most deeply problematic aspect of the cogito is, in my opinion, the dualism of mind and body that it inevitably seems to lead to. This dualism makes us ignore some very basic facts, such as the fact that linguistic meaning is established as a social practice, embedded in forms of life. Or the more simple fact, that Descartes could not have written his Meditations unless he sat, comfortable and warm enough, at a table, in a quiet and clean Dutch room, without being disturbed by screaming children or by screaming physical pain or by the fear of political or religious persecution.