Second cousin or half-second cousin?

Genealogy
1

AIO64-6.0.8–1, Win 11

Person X and Person Y share one common great-grandmother. The descent tree to Person X is via a first husband and the G grandmother. The descent tree to Person Y is via the marriage of a second husband and the same G grandmother.

My understanding is that second cousins share a set of two great-grandparents, and that half-second cousins share only one great-grandparent, rather than two.

Roots Magic identifies Person X and Person Y as half-second cousins.

On the other hand, Gramps’ Relationship Tool identifies Person X and Person Y as second cousins. FamilySearch also shows those two individuals as second cousins.

Does anyone know the correct answer?

2

I would argue that the half is correct on the basis that to be full
cousins the gt. Grandmother and gt. Grandfather would have to be the
same for both.
Think of it in terms of DNA handed down
phil