There are various beliefs in what/who is a Christian (the baptism definition is only for Catholics):
To conservative Protestants, a Christian is often defined according to their salvation status. Their definition is "true" to them, because it agrees with some of their foundational beliefs: that the Bible is inerrant, that salvation is by grace, and that one must be "born-again" to be saved and avoid eternal punishment in Hell
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To Roman Catholics, a Christian is often defined according to their baptism status. Their definition is "true" to them, because it agrees with their fundamental beliefs about the nature of sacrements, their understanding of the Bible, the declarations of many Church Councils, the statements of many popes, and their church's tradition.
To many in the very early Christian movement, a Christian was defined as a person who was baptized and proclaimed "Jesus is Lord." Their definition was "true" to them because it agreed with their understanding of their religious belief at a time when the Christian Scriptures (New Testament) had not yet been written and assembled.
And there are many others.
There is no consensus on what the "correct" definition of "Christian" is. There is only a near consensus within individual faith groups. Christians are those who regard themselves as being followers of Jesus even though they do not affiliate themselves with any particular religious group.
You can believe in God and not be Christian. Some Christian would say that Catholics are not Christian " I disagree. I think many do not believe Catholics are Christians because Catholics confess their sins to a priest (whereas most Christians confess directly to God) and Catholics believe in saints and many Christians believe this is worshipping idols.
And no, Christians do not have a monopoly on miracles. I would think that God would perform miracles on any person as we are all God’s children whether we believe or not. I heard many stories where those that are destitute and the worst people in society as far as morals and not Christians, are touched in some way by God. God will reach out to anyone who needs them even if they do not ask. A Christian should believe this especially where part of a Christian belief should have them to this.
Here is a great link that explains it, I think most thoroughly because it looks at different religions/faith groups.
http://www.religioustolerance.org/chr_defn3.htm