I agree, and the linked Wikipedia page starts out with the obvious definitions as well:

    In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle (the hypotenuse), and the cosine is the ratio of the length of the adjacent leg to that of the hypotenuse.

The author's claim that other definitions are the "most common starting points" seems like a straw man.