and i would expect cognates to be just as the person who was speaking to you did.. a family of words descend, descendant, descendancy, and so on. I am sure i would understand the word being used as you define it.. but that wouldn't be my first thought, or even my second.

the only other way i would think of cognates, would be a verb, especially an irregular verb.. i am, you are, he is, i was, they were...

(I went scurrying to the dictionary as i read your post.. and thought--Have i misunderstood this word for years? and while M-W 10th supports my usage, your meaning is listed as 3 a, and mine as 3 b and 3 c....)