The term “half-baptized” no doubt refers to these private ceremonies; it is defined in the O.E.D. “To baptize privately or without full rites, as a child in danger of death.”
A general notice to the people of Southea with Murrow in Cambridgeshire published in the Wisbech Deanery Magazine Dec 1890.
We should like to remind parents whose children are baptized privately, that they are not then half baptized, as it is commonly called, but truly and validly baptized, so that, as the Prayer Book says, if such a child should die before it commits actual sin it is undoubtedly saved; and that if such children " do afterwards live", they ought to be brought to the Church as soon as possible to be received as members of " the flock of true Christian people".
The Book of Common Prayer states about Private Baptisms
And let them not doubt, but that the Child so baptized is lawfully and sufficiently baptized, and ought not to be baptized again. Yet nevertheless if the Child, which is after this sort baptized, do afterward live, it is expedient that it be brought into the Church. If the Minister of the Parish baptized the Child, he may certify to the Congregation of the true Form of the Baptism, privately at such a time and such a place before witnesses.
If the Child were baptized by any other lawful Minister, then the Minister of the Parish shall examine those that bring the Child to the Church to determine whether the Child be lawfully baptized or no. If all things were done, as they ought to be, then he shall not baptise the Child again.
Stan