Пероги(perogies, pierogies) and Вареники(varenyky) are the same. But dumplings are the generalization for everything that's made of pieces of dough with meat (or any other) filling including Indian, Chinese, Japanese and other kind of dumplings such as gujia,modak,dango,mandu, jiaozi.
So perogies, pierogies should be accepted as well. Dumplings is not a proper way to say Varenyky.
I think it's a regional difference. Historically most of the Ukrainians who came to Canada and settled in the prairies in the late 1800's to the mid 1900's were from the Austro-Hungarian Empire and spoke a different dialect than people from farther east. My mother and her siblings made boiled dumplings filled with potatoes, and they were not called verenyky. My Anglophone ears heard the word as "pedokheh". It was probably either пероги or пироги.