"I walk with my cat on Sundays."
Translation:Siúlaim le mo chat ar an Domhnach.
"On Sundays" here means habitually, so you would use Ar an Domhnach. You would never use domhaigh with the ar an structure. However, if you are talking about a particular Sunday (usually the one coming, ex "I will walk my dog Sunday") and not habitually, you would use Dé Domhnaigh.
Siúilfaidh mé mo mhadra Dé Domhnaigh (I will walk my dog Sunday)
I didn't say "Yes, days of the week are always definite in Irish" because I don't know that to be true. Maybe it is, maybe it isn't.
I didn't say that you were "being English", I said that were seeing things from an English speaker's point of view. I don't need to know whether "days of the week are always definite in Irish" to recognize the chauvinism in your assumption that Irish must be wrong because of how you analyzed the sentence in English. If you don't find it useful to be reminded that when Irish and English take a different approach, it isn't because there's a flaw in the way Irish works, then so be it.