If you're doing stuff with point sets, "unit circle" typically refers to the point set satisfying x^2 + y^2 =1, the "unit disc" or "unit ball" would be used to

If you're doing stuff with point sets, "unit circle" typically refers to the point set satisfying x^2 + y^2 =1, the "unit disc" or "unit ball" would be used to refer to the point set satisfying x^2 + y^2 < 1 (open unit ball) x^2 + y^2 <= 1 (closed unit ball).