According to one standard theological paradigm, a Christian healer is someone who can miraculously cure medical conditions in general.
But according to the Gospels, an exorcist can be a healer. The Gospels distinguish between natural medical conditions and demonic medical conditions. Some medical conditions have natural causes while other medical conditions have demonic causes. Possession can manifest itself in medical conditions.
In addition, Ezk 13:17-23 seems to indicate that it's possible to cause a medical condition by hexing the victim. Both possession and witchcraft can result in some medical conditions.
That, however, complicates the analysis of miraculous healing. In principle, an exorcist could cure someone of a medical condition that's caused by demonic activity (i.e. possession, witchcraft), but be unable to cure someone of a medical condition that's caused by natural factors.
Moreover, there is no gift of exorcism. The ability to cure medical conditions in that situation is indirect. An exorcist doesn't have the power to simply heal someone of their medical condition. At most, he has the power to break the occult bondage that's causing the medical condition.
Furthermore, exorcism is really a matter of invoking God's mercy and power. It's not really an ability on the part of the exorcist.
This, in turn, imposes a potential limitation on a Christian healer–assuming that's an accurate classification to begin with (see below). What if Christian healers can only cure medical conditions that have occultic causes rather than natural causes?
There's also the question of whether there's such a thing as a Christian healer. In the locus classicus (1 Cor 12:9), Paul's usage is ambiguous. He doesn't say a "gift of healing" but "gifts of healings."
So he may not even mean that some Christians have a gift of healing. Rather, every healing is a gift from God.
That's not to deny that some Christians might be agents of God's healing power, but this might be intermittent or unrepeatable. Say, on one occasion, God grants a Christian mother the ability to lay hands on her deathly ill child and convey healing. That might be a once-in-a-lifetime event.
On that interpretation, both the charismatic and cessationist paradigms are defective.