Interesting. Reincarnation without a supernatural explanation. Is that a thing?
There's lots of blending that goes on with these terms. There are Atheist Buddhists who don't believe that they are going to be reborn as a dog or something, but instead believe that treating others as if they were yourself in a different life is simply the best and most ethical way to go about living. There are Buddhists who call themselves Atheists because they don't believe in any specific deity, but who do believe in supernatural entities/processes. They believe that they will be reborn into a life befitting their karma. They are Atheists because they don't believe in a God, but some other people who call themselves atheists would say that they are not actually atheists because they believe in supernatural beings/processes.