I think this relates to the
Burial Laws Amendment Act 1880.
This defines the administrative steps for getting someone buried in an Anglican church graveyard without an Anglican ceremony. First, "
any relative, friend, or legal representative having the charge of or being responsible for the burial of a deceased person may give forty-eight hours notice in writing" to the minister or whoever else does this "
that it is intended that such deceased person shall be buried within the churchyard or graveyard of such parish or ecclesiastical district or place without the performance, in the manner prescribed by law, of the service for the burial of the dead according to the rites of the Church of England, and after receiving such notice no rector, vicar, incumbent, or officiating minister shall be liable to any censure or penalty..."
There is a form for that notice, and another one to fill in afterwards which is often called a certificate of burial, and goes to the parish so the burial register can be filled in. Presumably it comes to the registrar as part of a copy of those parish register entries. I'd guess the person who fills it in is usually the undertaker, like today, though maybe it could be an executor. Of course there was (and is) also a certificate that goes the other way from the registrar (or else a coroner's order) to authorise the burial.
While the act says a non-Anglican burial can be done like that, I don't think it says it has to be. But that's not entirely clear, and there are several sections that apply to "any burial under this Act", which probably ought to still apply. So if the RC priest just had a word with his Anglican brother (they likely knew each other well) I don't think the forms are required, and that may have happened in the second case. If the burial register is filled in, the certificate has no purpose in the Act, and in any case it's not stated in the Act that the registrar needs to know about it.
Any burial obviously has to be arranged with the people who run the graveyard - allocate a grave, dig it, arrange access, etc. The act says only a little about all that, but does insist that those arrangement should be the same as usual.