This is an interesting topic for me because I experienced so much variation with position during my genital exams -- all at the same practice (mid-80s through the 90s in the northeast US). During the time that I went there, I saw three different doctors (I'd just get an appointment with whichever one was available, so which one I saw was somewhat random).
One of the doctors, who I only saw twice (at about 14 and 18) used the most common position of having me standing, underwear lowered, and he was seated in front of me.
Another doctor, who I saw throughout my time there, would have me standing, but he'd stand behind me, reaching around to check my genitals and test for a hernia.
The third doctor at the practice, who I saw most frequently (and saw last at age 17), would do both the genital and hernia exam with me laying flat on the table. Once in that position, he'd pull my underwear down rather nonchalantly and get right to the exam.