In class we had some proposals for the definition of rectangle. Here are some of them.
(1) A quadrilateral is a rectangle if it is a parallelogram with four right angles.
(2) A quadrilateral is a rectangle if it is a parallelogram with one right angle.
(3) A quadrilateral is a rectangle if it is a quadrilateral with four right angles.
There are other, odder options, such as this
(4) A quadrilateral is a rectangle if it is a parallelogram with congruent diagonals.
Definition (1) is a natural, civilized definition. It is descriptive and it is also the definition in BG. In effect, this defines a rectangle as a special kind of parallelogram, which has a lot of merit.
However, in math we often try to give minimal definitions with the fewest number of things to check. Either definitions (2) or (3) require fewer points to check than does definition (1); both imply (1) by the properties of parallelograms.
One merit of definition (3) is that it echoes the meaning of the word "rectangle" which means "right angle".
Definition (4) is also equivalent, but it does not seem a very natural match to the way we think about rectangles.
In the end, since we can prove 1, 2 or 3 equivalent from the properties of parallelograms, we will not object to any of these 3 definitions in proofs. So just choose the one you like, but please be consistent within a single proof.
Note on spherical quadrilaterals: Since there are no parallel great circles on the sphere, the only definition among these 4 that makes any sense is (3). However, the angles will not be right angles on the sphere, so we conclude that there are no rectangles on the sphere.