A kite is a quadrilateral with two pairs of adjacent congruent sides. The angles between the unequal sides are equal.
A General Quadrilateral
You have described a rectangle.
This is a rectangle.
A rectangle (that is not a square).
your mom, and your cousin
This is a parallelogram. The first requirement is 2 pairs of congruent sides where the congruent sides are not adjacent. This is like a rectangle (excluding a square) that has two pairs of congruent sides where the congruent sides are not adjacent. But the angles are not all congruent (as set in the question) which pushes the shape into the "next less regular" shape, the parallelogram. The angles will not all be congruent, but it will have 2 pairs of congruent angles. There is no way to avoid the 2 pairs of congruent angles because of the requirement that the shape must have 2 pairs of congruent sides (the first requirement).
An isosceles trapezoid
Two pairs of adjacent sides are congruent. The angles between the non-congruent sides are congruent.