Why was belief in God not defined in the aforementioned Prophetic Hadith, and yet we find other definitions for it?

There is no contradiction between the two definitions of faith. The Prophetic hadith that defines faith as bearing witness that there is no god but Allah necessitates belief in the existence and Lordship of Allah. Bearing witness that Muhammad is the Messenger of Allah is a fundamental pillar because whoever believes in Allah will affirm His Messenger. Likewise, actions performed with the limbs, such as prayer and zakat, are evidence of the sincerity of the inner faith. Faith is not by adornment or wishful thinking, but it is what is firmly established in the heart and affirmed by actions. Therefore, the definition of faith mentioned by the scholars is derived from the hadith and other religious texts, and there is no contradiction between them.