The only condition is that the document must be within its validity period for the purpose of proving citizenship, which includes the validity written on the document but can go beyond it. There's also an implicit condition that the document must not have been revoked (because it was declared stolen, or because it was found to have been obtained fraudulently): if it's been revoked, it's deemed not to exist anymore. An additional subtlety is that French law cannot dictate what other countries consider to be sufficient evidence of French citizenship.
The wording on the website of the embassy in Thailand is strange and I couldn't find similar wording elsewhere. For example, the general consulate in Buenos Aires states
La carte nationale d’identité est un document officiel qui permet à tout citoyen de justifier de son identité et de sa nationalité française.
i.e. ”the national identity card is an official document which allows any citizen to prove their identity and their French nationality“.
The relevant law when you need to prove your French citizenship to a French government service is article R113-5 of the Code des relations entre le public et l'administration (code governing interactions between French government services and the public). It states that a “carte nationale d'identité en cours de validité” (identity card within its validity period) is sufficient proof of identity and citizenship. An official government website states that the document remains valid as a proof of citizenship for 5 years after its expiration date. I can't find the law that codifies this (for the specific case of renewing an identity card, it's in article 4-1 of decree 55-1397, and for the deliverance of a passport, in article 5 of decree 2005-1726).
My understanding is that this law does not cover border crossings. I could not find the French laws that cover crossing a French border as a French citizen.
A number of countries, including all European Union and Schengen Area countries, accept French identity cards as a proof of French citizenship, granting the right to cross their borders. These documents are listed in the Public Register of Authentic identity and travel Documents Online. The entries for French identity cards do not mention any conditions other than the validity period.
A subtlety about the validity period of French identity cards is that cards issued between 02/01/2004 and 31/12/2013 (except to minors) had their validity extended after issuance: the validity period shown on the card is 10 years, but a subsequent law extended the validity to 15 years. Although PRADO mentions the 5-year extension, not all countries accept it. The French government publishes a list of countries according to whether they do or do not accept it.