Having done quite a lot of digging into baptism practices in the Church of England in the late eighteenth and early nineteenth century, here is what I think may have happened, and why, and what mysteries still remain (with thanks to everyone who has commented so far).
Many thanks to AndyW, AdrianB38 and Sempaiscuba for helping me work out what was going on.
The (possible) sequence of events
Maria Ann Weal (who was John Charles Wright's [henceforth JCW] mother) was born in Kidderminster and was either living there or had returned there when she gave birth to her first child (it wasn't unusual for a woman to go to her mother at this time, and Maria Ann was only 19 in 1830.) Her husband John Wright was described as 'of Kidderminster' at their marriage in 1828 and 'late of Kidderminster and formerly of Wolverhampton' upon his death in October 1830; he was buried in Wolverhampton.
JCW was privately baptised in 1830 in the vestry of St Mary's Kidderminster by the Rev. Thos. Cooke who had been curate there for 22 years before his death in 1837: Gentleman's Magazine of 1837. Confirmation that this ceremony took place in Kidderminster comes from the fact that one of the witnesses named (who had a very distinctive name and occupation) was definitely resident in Kidderminster from 1829 to 1841 (Trade Directories and Census); this witness died in 1841. His parents were issued with some sort of certificate that the baptism had taken place (perhaps because they already intended to carry out the next step in the process and having the baptism certified was helpful in that) but more likely because a certificate of baptism could be used as proof of identity before civil registration was introduced.
Later, JCW was publicly received into the church and his baptism entered into the register of St. Peter's Wolverhampton (the only parish church in Wolverhampton at the time). His parent's abode was given as Wolverhampton.
Why the private baptism?
This is open to conjecture. The Book of Common Prayer makes provision for private baptism when need compels (the infant is likely to die before the next Sunday when a normal baptism would be carried out) but there was also a tendency for certain parts of society to prefer the privacy of a ceremony carried out not under the public gaze. As this ceremony took place in the church, it seems unlikely that the child was too ill to leave the house... but having the child 'received' in Wolverhampton suggests that the public gaze wasn't an issue.
Perhaps because the parents were living or about to live in Wolverhampton, they wanted JCW to appear in the register there -- as proof of his legal 'settlement' there under the Poor Law rules? (See https://genealogy.stackexchange.com/a/3485/6485 for more on this). Or perhaps the godparents they wanted for their child were in Wolverhampton? Or were they worried about the child's health, or their own health (John died a few months after the baptisms, and Maria Ann died in 1834 after a long illness).
It may be that I will never know the reason... As the original document is lost, and Worcestershire Archives do not have any papers deposited from the solicitors who were involved in the property transaction in the mid 1860s that necessitated a copy of the document... those solicitors still have the same office in Kidderminster in which they have operated since 1828 and it has a deep cellar and they sent a member of staff to have a look, unfortunately without result but kudos to them for trying!
The Clergyman's Manual by Robert Simpson (142) has a few things to say about private baptism! Notably:
Baptisms in the vestry are irregular. No baptism in a church is
regular without there being godfathers and godmothers present,
according to the rubric. Parents are not allowed to stand as sponsors
for their children; nor are persons not communicants qualified for
The distinction between public and private baptism does not
consist in the one being administered in the church and the other in
a private house; but in the one taking place in the great
congregation, where the child is publicly received amidst the prayers
of multitudes, into the congregation of Christ's flock; while the
other is performed privately, before three or four witnesses,
sometimes at the font, sometimes in the vestry, occasionally at the
communion table; nor have I ever been able to discover any reason for
making use of the service for the public baptism of infants in an
empty church, which would not apply to the adoption of it before a
large party in a drawing-room. (Rev. T. Webster.)
[I think we can assume that Robert Simpson was not too enamoured of private baptisms except when necessity compelled...]
The Book of Common Prayer makes no mention of godparents being required at a private baptism. Godparents are required at a public baptism, and also at the point where a child that was privately baptised is subsequently brought into the church to certify publicly to the congregation that a 'lawfull and sufficient' baptism had already taken place. There is provision for a child to be received by a minister other than the one that carried out the private baptism, which means that the two events could happen in different parishes.
The private baptism was NOT entered into the Parish Registers of st Mary's Kidderminster. The entries in parish registers for "private baptisms" actually refer to the event in which the infant is 'received into the congregation' -- as discussed at length in Few Deaths before Baptism: Clerical Policy, Private
Baptism and the Registration of Births in Georgian
Westminster: a Paradox Resolved which argues that:
The evident lengthening of the interval between birth and baptism over
the eighteenth century has often been assumed to have increased the
risk that young infants died before baptism. Using burial records that
include burials of unbaptised infants and give age at death we
demonstrate that very few infants who survived the first few days of
life escaped baptism in the parish of St. Martin in the Fields,
despite a very profound lengthening of the delay between birth and
baptism over the second half of the eighteenth century. Examination of
baptism fee books indicates that perhaps a third of all infants were
baptized privately in the parish and a pamphlet dispute between the
vicar and one of his clerks provides extraordinary evidence of the
extent to which baptism was a process rather than a single event. Our
analysis suggests that it was the registration of baptism that was
delayed, with no affect on the risk of death before baptism.
and refers to The Population History of England, 1541-1871 by E. A. Wrigley and R. S. Schofield page 96-97 (various editions) which says:
even in parishes where home baptism was widely practiced such baptisms
were often not entered in the register unless the child lived long
enough to be received into the church by public ceremony