When a family letter, newspaper reference, interview or other item refers to two individuals as being “cousins,” remember that the relationship may not be a “first cousin” (an individual with whom a person shares a set of grandparents. The reference could be to one of a variety of relationships.
In informal writing, I use the word “cousin to refer to a relative with whom I share an ancestor and who is not my own direct-line ancestor or an aunt or uncle. The person may be a second cousin, third cousin, fourth cousin once removed, etc. I’m not the only one who does this and your grandma’s reference to someone as her cousin may mean what my reference to the word does or it may even indicate other relationships–including ones by marriage.
In non-genealogy writing to clarify the relationship somewhat, I may refer to someone as my “Rampley cousin,” “Ufkes cousin,” etc. Another approach is to indicate the relationship a little more specifically–cousin of my Grandma Neill, Great-grandma Habben, etc. That way someone has a sense of whether the cousin is related to them or not and how they are related to me. I avoid the “cousin/removed” statements where possible and indicate the relationship if more clarity seems necessary–“his grandma Henerhoff and my grandma Neill were first cousins,” “our Mom’s mom’s were sisters,” etc. (and give more detail if necessary).
But when you see the word “cousin,” don’t assume the two individuals shared a grandparent–the connection may have been a biological one further back, could have been a relationship resulting from a marriage, or even a family friend who was considered a relative.