Puzzleannie has it about right. The person in Israel was acting upon a shallow (if even existent!) idea of baptism. The sick lady was expressing the natural desire for some rite that might help her make her peace with God at the hour of her death–but she was acting contrary to what she’d been taught all of her Protestant life.
Many Protestants, like many Catholics, actually practice a “folk religion” made up of bits and pieces they’ve picked up here and there. Either their church doesn’t stress education, or they’re not there enough to hear the teaching, or they really have no commitment to truth and prefer to live a homespun religion that makes few demands outside their comfort zone.
That said, there is a subspecies of Baptists who believe that baptism is the initiatory rite into a local church. I’ve never actually met one of these people, but they are hard shell individualists. When you move to another such Baptist church, you get baptized into that one, and so on.
For the rest of Baptists, and Protestants of all stripes, one can only be truly baptized once. Since Baptists stress that baptism must come after conversion, where one passes from death to life (John 5:24), some Baptists, over a period of years, wind up going through the ritual three or four times until they decide that their conversion is finally real. The last time, however, is the only one that they will call “baptism.” The other times, they say, they just got wet.
The original name for Baptists was “Anabaptists,” the Greek prefix “ana” meaning “again.” They rebaptized those who had been baptized in infancy. Of course, since they didn’t believe that the infant ritual was true baptism, they rejected the label “ana.”
Likewise, they don’t believe that sprinkling qualifies as baptism. So if someone wants to join a Baptist church, but was previously only baptized by sprinkling, he would have to undergo a “proper” baptism.
The Churches of Christ believe that baptism is the God-ordained way of receiving the grace of salvation, but it must be administered in the Bible way–and since they’re the only ones who really obey the Bible, only their baptism is valid. So you’d have to be rebaptized to join one of their churches. Conversely, many conservative Baptist churches won’t recognize a baptism done in a “Church of Christ” because of the baptismal regeneration doctrine–so one of those would have to be rebaptized to join such a Baptist church (as Puzzleannie pointed out above). But not all Baptists are quite that exclusive.
Yeah, it gets kinda crazy over here sometimes, but we still have a good time.