About the surname, according to MacLysaght's Surnames of Ireland, the original Irish form was Mac an Tiompánaigh, meaning "son of the tympanist/musician". It originated in County Down, where it was anglicized as (Mac) Timpany, Atamney, Tameny, Tempany, and even Tenpenny. The other variations you listed are not mentioned, but are logical attempts at anglicization, depending on who was making the attempt (non-Irish speakers often had/have trouble hearing some of the sounds of the Irish language). He mentions that the name is now rare in Down, but can also now be found in Derry (usually as MacAtamney).
As to the first name discrepancies, Eugene was a common anglicization of the Irish name Eoghan, which is actually an old form of the name John, based on the original Hebrew name (other forms of John being Seán, which came with the Normans, and forms like Ewan, Iain or Ian (which were more commmon in Scotland). If the family named a child Eoghan, the parish priest may have insisted on "Latinizing" it to Eugene or something like Eugenius in the parish register, although the family may never have called him that. Later on, he may have chosen to use John instead, so I wouldn't necessarily rule out the John Tameny whom you found. His wife, Sally Hendren, had a Norman surname which is found mostly in Armagh, according to MacLysaght, though the usual spelling is apparently Hendron (sometimes it became Henderson).
The same is true of the names Sally and Sarah, which were both used as anglicizations of the Irish name Sorcha. The family may have named the child Sorcha, but the priest may have insisted on using Sarah in the parish register, because he thought it was Latin (it's actually Hebrew, of course). She may later have chosen to use Sally when she grew up, and it's entirely possible that she was never actualy called Sarah by anyone but the priest (although sometimes Latinized parish regster names did "stick").
You may already be aware of this, but there are records in the Family Search database [https://www.familysearch.org] for both families which you mentioned: Eugene McAtamney at MR63-8LZ and John Tameny at GQHY-CFZ. Eugene appears to have married a second time, after his first wife died, to a much younger woman, and appears to have had a total of 16 children, but none named Sarah. Given that both families lived in Seagoe at the same time, they were probably related, so perhaps someone has mixed up some of the info. At Family Search, you can see the screen name of each person who creates or modifies a record, and can send them a private message, so if you weren't already aware of those records, you might try contacting the people who input them and compare notes.
