Mr Griffith is the preacher at the local Chapel in the village where Daphne and her family live. As was the case in Victorian times when there was a greater religious tolerance (compared to the horrors of previous centuries) and the first faint stirrings of ecumenism (ie, a civilized common-sense understanding that the various denominations within Christianity had more in common to agree about, as opposed to the doctrinal differences that separated them) he was graciously invited to preach the sermon at the local Anglican church one Sunday.
This left an indelible mark on Daphne, and even her grandmother could only say "Well!" afterwards. To Daphne, a good hour of fire and brimstone from somebody who really believed was like being dunked in ice water, and challenged as to what she really believed in. This was her first exposure to a priest of a different religion, and stood her in good stead for the Nation.
By context and inference, Mr Griffith has a lot in common with strict-observance Llamedosian druids on the Discworld: the name, the fiery impassioned belief, and the ability to spray a fine mist of spit as he preaches, are all a dead giveaway.
The social hierarchy of the Church in nineteenth-century Britain needs a few words. At the top was the state church, the Anglican (Episcopalian), where the ruling monarch was the head and Defender of the Faith. (a tradition handed down from Henry VIII). Anyone who was anyone would have gone to church here on a Sunday. Daphne's grandmother would have insisted on her social status being observed, right down to having a pew at the front as was her right, as somebody standing maybe 140th in line to the throne. The rest of the landed gentry would have taken progressively lesser seats, with peasants at the back - well, they couldn't actually be forbidden from attending, that would have been unchristian. You just had to tolerate them with good grace.
The various Nonconformist, Methodist, Presbyterian and Chapel denominations were seen as the social preserve of the landless city workers, farm labourers, the poor, the dispossessed and the feckless. Apart from a few eccentrics and mavericks, nobody from the educated upper classes would dream of worshipping in such socially unfavoured conditions. During this period, Wales went through several "religious revivals", and the llasting llegacy was a rash of Chapels across the country, as well as a thriving export trade in preachers. Many of the great hymns are also a product of the 19th century, such as Cwm Rhondda/ Bread of Heaven and A'r Hyd a'r Nos, are products of the Revival.
And of course Roman Catholics remained ENTIRELY off the social scale: apart from a few English mavericks in remote places like Norfolk and the Scottish borders who weren't being actively persecuted any more, this was a lesser religion for the immigrant Irish, a damned nuisance, but nothing we can do in that infernal country can shake them away from their dependence on Romanism, and anyway we need people to dig our canals and build our railways.
Daphne's grandmother may not have liked Mr Griffiths preaching at her church, but she'd have been really up in arms at the thought of a Roman Catholic priest being offered pulpit time!