Does a person's statement, "I am a Christian by my faith," mean that Islam is an extension of Christianity, and that original Christianity was distorted, and that a Muslim is more Christian than Christians themselves?

The name Allah chose for His servants is Islam, as stated in His saying: ﴿He named you Muslims from before and in this.﴾ It is not permissible for a Muslim to describe himself as a Christian or a Jew, because these descriptions have come to denote the worship of other than Allah or disbelief in the Messengers, and this is forbidden and may reach the level of disbelief. The Sharia (definition) texts have warned against naming oneself with other than Islam. So, whoever swears by a religion other than Islam falsely, he is as it was said. The religion of all prophets, including Moses and Jesus, is Islam, which is submission to Allah through monotheism and obedience. A Muslim must be wary of words that may lead to Allah's displeasure, and of affectation in speech and action, and adhere to the guidance of the believers in naming themselves and attributing themselves to the religion of Islam.