I'm not too familiar with archer and marksman. And archer points to different moves for different people, so that's a confusing one anyway.
To me, marksman is the inverted grip with the inside hand up top, arm stretched, and the outside arm bent, hand right about shoulder height. Use of legs may vary.
To me, bow and arrow is any inverted move with your back to the pole, both legs straight, one straight up, with the foot hooked, the other pointing forward, away from the pole. Many different grips/stances can be used to perform bow and arrow, including elbow, head or handstand, iguana/pencil grip, marksman grip, etc.
I'm staying clear of archer, because of the different definitions that are floating around.
Anyway, that's just my take on it. There are surely many different interpretations of these concepts out there!