Yes, both the sun and moon rise in the EAST and set in the WEST.
The moon happens to rise in a place that lines right up with a nice up hill gallop section of our trails around the ranch
Sun up and sun down are easy to notice - but the moon will rise during the day sometimes (and hardly noticeable due to the day light!), like I mentioned, the rise time can jump as much as 45 mins between days.
And rise time is just that - when it crests the horizon. If using moon light for riding, its best when the moon rises a few hours before you are going to be out (as its then high in the sky).
And I much prefer riding in the dark - rather than using headlamps. I find the headlamps really make it so my eyes can’t adjust, and I can’t SEE. I also wonder how it messes with the horse’s vision - their eyes being to the sides and all - seems like it would be easy to blind them with the beam.
This week I have been trail riding before the moon as risen - Once my eyes adjust I am fine - and horse is 100% just fine in the dark.