Is 'Rebirth' in Buddhism something different from 'reincarnation'?

Growing up, I had a general understanding—based on lay textbooks and common interpretations that the dharmic religions like Hinduism, Buddhism, and Jainism all believe in reincarnation: the idea that an individual is reborn into a new body, either human or animal, after physical death of the body. As I became more interested in Buddhism and tried exploring it more deeply, I noticed that different Buddhists seem to explain this concept in very different ways. Some use the word "reincarnation" and appear to mean it quite literally, while others insist that "rebirth" in Buddhism is not the same as reincarnation, especially since Buddhism denies the existence of a permanent self. This has left me quite confused. Is rebirth just another term for reincarnation, or does Buddhism teach something fundamentally different? What is the correct way to understand the concept of rebirth in Buddhist teachings?