Ancestry/Genealogy

they have transcriptions and images for many parish registers.

There appear to be quite a few more possibles in the area than those you listed, too; a quick search returns:

Mary Morgan, chr. 4 Apr 1813, Llanbadarn Trefeglwys, Cardiganshire, father, David, mother, Anne

Mary Morgan, chr. 5 Apr 1816, Llanbadarnfawr, Cardiganshire, father, David, mother, Jane

Those would all be christening records; birth records weren't kept until some time later, in most cases. Those are all from parish records, in most cases it's safe to assume the actual birthdate is some six weeks before the date of christening.

British records are very indifferent in some cases, because prior to say the 1800's most of the population would have been agricultural labourers in rural areas. With the Industrial Revolution and urbanisation and increasing literacy, better records were kept, but in most cases the furthest back you can expect to trace someone in Britain is back to the time of the earliest parish records (which in some cases go back as far as the 1500's). This is somewhat complicated by the fact that parish records are the only record for most places, and they were not infrequently lost due to fires (or destroyed in the Civil War...there cna be significant gaps for the 1650's), or, in some cases, the incumbent vicar refuses to release the parish register for transcription. And in some cases records just simply weren't kept up very well (except for the upper classes).

And on edit, here (parish records search at findmypast.co.uk): http://www.findmypast.co.uk/search/parish-records/baptisms