For the conditions to see a part of a rainbow, I would say you need the following:
- sufficient moisture to form numerous micro spheres of water, which creates the refraction pattern
- the right angle between the observer and the sun
- sufficient mass-thickness with this angle and with this composition to build up a visible amount of color
Since the sun is practically infinitely far away, the distance between you and a drop of water doesn't really affect the angle. Because of that, you can have a coherent rainbow shape form, because one line of sight will consistently reflect red, blue, or whatever wavelength.
However, looking up or horizontally builds up potentially miles of a line of sight. If I wanted to be pedantic, I would claim that literally all rainbows you see are 360 rainbows. The parts below the horizon are just too dim for you to make our with your eyes because they are not high enough sensitivity, nor do they have the ability to make out those colors against the brighter background. To drive home my point, consider such an image from an aircraft:
https://i.stack.imgur.com/TaYXC.jpg
I hope you find it somewhat believable that the rainbow is less bright as we look at the lowest part of it. This part is formed by the atmosphere between the aircraft and the ground. Because of that, the distance of this line of sight is limited. Perhaps a few 100 feet, it depends on how high you're flying. Compare to a human standing normally. The line of sight to the ground may only be a few feet. This is not enough to intersect with enough water droplets to see the pattern.