Well, I that's what you meant, then fine - I usually use 'the cogito' as shorthand for the entire expression, 'cogito ergo sum' (and the thought it represents). Given that cogito is literally a first person singular verb, I think it might be useful to use a different description to indicate the impersonal 'there's thinking going on' - but that's just semantics at this point:)

Not even the cogito has been established with agreement between philosophers. As far as I can tell, the part everyone agrees on is something like 'thinking is going on', but involving the terms 'I' and 'being' is contested.