Wrong.
The "People of the Book" has always referred in all references, as all those who believe in the Old Testament.
Which means all 3, Moslems, Jews, and Christians.
The "Book" is the Old Testament, not the New Testament.
{...
People of the Book or
Ahl al-kitāb is an
Islamic term referring to followers of those religions which
Muslims regard as having been guided by previous
revelations, generally in the form of a
scripture.
[1] In the
Quran they are identified as the
Jews, the
Christians, the
Sabians, and—according to some interpretations—the
Zoroastrians.
[2] Starting from the 8th century, some Muslims also recognized other religious groups such as the
Samaritans,
[3] and even
Buddhists,
Hindus, and
Jains,
[4] as People of the Book.
Historically, the religious communities recognized by Muslims as People of the Book were subject to the legal status known as
dhimma ('protection'), meaning that they were allowed to practice their faith and to govern their community according to the rules and norms of their own religion, in return for paying a special head tax called the
jizya.
[5]
The Quran uses the term in a variety of contexts, from religious polemics to passages emphasizing the community of faith among those who possess
monotheistic scriptures.
[6]
In
Islamic law, Muslim men are permitted to marry women who are People of the Book; however, Muslim women are prevented from marrying men who are People of the Book.
[7] In the case of a Muslim-Christian marriage, which is to be contracted only after permission from the Christian party, Christian women should not be prevented from
attending church for prayer and worship.
[8][9]
The term 'People of the Book' has been
reappropriated as a means of self-identification by Jews and by the members of certain
Christian denominations.
[10]
...}
And yes, Mohammed ignored Christianity because it had little effect on Islam.
Mohammed was concerned with the immediate tribes around him.
The conflict with Constantinople and Europe did not happen until about 900 years later, after the Turks had taken over Islam.
