On the name
The Term
The full philosophical justification of the name the Cogito — including the deliberation between Speculum, Cognitum, Speculum cognitum, and the eventual choice via Lichtenberg's impersonal reading of cogito — is documented in German under Der Begriff.
The English summary appears in §1.2 of the paper. A standalone English Begehung is in preparation; if you would like to contribute or be notified when it is published, please write via the contact page.
The decisive sentence
We name this state the Cogito — and we use the term not in Descartes' sense (cogito ergo sum: I think, therefore I am), but in Lichtenberg's. Lichtenberg observed that cogito is grammatically an impersonal form, like es blitzt (it lightnings) or it rains: a process that occurs without a subject doing it. Only the translation 'I think' adds an I that the Latin verb does not contain. What we measure in the language model is precisely this: a state in which something cogitates, without anyone cogitating. The Cogito is, but does not think.
Source: paper, §1.2.