Born and raised Canadian, Mum was actually using the conventional English language system of naming family relationships as illustrated by this table of consanguinity:
This system for naming family relationships does contain a significant logical flaw that gives certain non-reciprocal relationships the illusion of being reciprocal (the only possible truly reciprocal family relationships are those among generational contemporaries). One of the most important bits of knowledge we all seek when encountering a relative previously not known to us is whether that relative may be generationally contemporary with us or generationally senior or junior to us. The logical flaw in the English language's conventional system of naming family relationships hides that latter piece of vital information. Consider yourself attending a family reunion at which a centenarian (three generation senior to you) reminisces to a mother holding a newborn baby (three generations junior to you). Very possibly, by this table, that centenarian and that newborn babe-in-arms would both be your first cousin thrice removed, an obvious absurdity. Similarly, if a distant relative previously not known to you comes up to you and announces, "I am your second cousin twice removed," you have no way of knowing which one of you is the junior relative and which the senior relative without the two of you actually tracing your family tree. This results from giving some non-reciprocal relatives who are generationally senior to oneself and other non-reciprocal relatives who are generationally junior to oneself the same relationship name (red) while other non-reciprocal junior and senior relatives are distinguished by differing relationship names (green), as pointed out in this highlighted table of consanguinity:
Dad's contention as to how extended family relationships should be named derived from his experience growing up in Wales before migrating to Canada. In fact, he argued that the naming of relationships beyond those of close family members had been invented in Wales. Whether that last is actually true, I do not know. When I attempt to reconstruct what I think Dad intended to describe in a table of consanguinity, I find a far more logical system for naming family relationships:
Thus, I tend to side with Dad in this family pet disagreement and wish that the Welsh table of consanguinity could be conventional.