Is every Muslim a believer, or is it the other way around?

Can one be a Muslim without being called a believer?
There is scholarly disagreement among the people of the Sunnah and the community; because the Muslim is broader than the believer, and the believer is more specific. So, if the term Muslim is used in a general sense, it encompasses everyone: the Muslims, the believers, and those who excel in worship. However, a Muslim may be described solely by the attribute of Islam, without the attribute of faith, in the context of praise, if he is among the sinners. It is said: he is a Muslim, but not a believer with complete faith. However, it is better in such a case to say: a believer with deficient faith, or a believer who is an open sinner, or a believer in his faith, but an open sinner due to his major sin, as Abu al-Abbas mentioned in "Al-Aqidah al-Wasitiyyah"; because the station of praise with faith is a great station. And Allah detailed the distinction between them, saying: [Al-Ahzab:35]. This is in the context of detailed specification: "The adulterer does not commit adultery while he is a believer," meaning: rather, he is a Muslim, not a believer with complete faith—the faith that its holder deserves full praise. Therefore, it is necessary to reconcile the texts in a way that harmonizes them and clarifies their meaning.