My understanding of the cycles of life, death, return to life, ad infinitum, is that each has elements of both your definitions of rebirth and reincarnation. It is our souls choices as to what vessel we will incarnate, but that those choices are also driven by what is needed, not just for the individual but also for each soul in the group that we tend to travel with. https://medium.com/illumination-curated/life-is-school-for-the-soul-9b081dbe8453