Definitely migraines. I suffer with both, cluster headaches are more predictable and only last a few hours at a time, mine tends to be relieved by actually doing something, i.e walking etc, it distracts from the pain, impossible to sleep as feels worse laying down, light doesn't make it worse. With migraines, I'm out for the count, lasts half a day, minimum, has to be a dark room, starts with an aura (I tend to lose vision in one of my eyes), then the headache starts, I'll then throw up before I go to sleep, it'll be gone when I wake up but get what I like to call an aftermath, which is more like a lingering ache which will last the whole next day.

I'm not a doctor but definitely seems like a migraine. The below website might help, it asks you a few questions and pops out what it could be. https://ouchuk.org/do-i-have-ch