Edward and Edmond are variations of the same name, so I'd always search for both.
Even though the date his gives for his birth is the same on those two records, I'l treat with caution - ages and dates are often inaccurate, also when searching I'd leave out the middle names, as these often dont appear on births or baptisms etc (I dont think his CR10 merchant navy form had a middle name).
The BMD Index lists names of registration districts, so for Co. Waterford the ones to check are : Carrick-On-Suir *, Clogheen *, Clonmel *, Dungarvan, Kilmacthomas, Lismore, Waterford * (city) and Youghal *
Those with * covers parts of other counties.
Maybe your William was born in an area close to the Waterford Co. border, or the family lived for a time in the City of County of Waterford after he was born ?
I think the first clue to follow up on is the father - I'll do another search for focusing on the occupation..
Where in the US did Edward/Edmund jnr die, have you found a death cert and/or obit. or headstone ? (US certs often include names of both parents, and obit. or headstone might show a more detailed place of birth)
If the date of birth he gave is accurate then this index record is defintely worth following up, right year/quatrter, and the district covers parts of Co. Tipperary and Co. Waterford :
Name : Edward Hayes
Event Type : Birth
Quarter and Year: Apr - Jun 1896
Registration District: Carrick-On-Suir
Volume :4 / Page :484
I'll see if that Hayes family appears in Carrick area in 1901/1911 and see if it's possible to find the father's name.
p.s. there is a railway line and station close to Carrick on Suir