Short answers, pulled from the story.
The Buddhist path to liberation is called marga. It is described in many different ways across Buddhist traditions, with the Noble Eightfold Path being the most widely known formulation, though the Sutta Pitaka preserves several other summaries.
The four dhyanas are four progressive stages of meditative absorption described in early Buddhist texts. Each stage involves a deeper settling of awareness; the fourth jhana serves as a basis for three further kinds of knowledge, culminating in the destruction of the asavas and realization of the Four Noble Truths.
The Seven Purifications are the classical outline of the Theravada path, as described by Buddhaghosa in the Visuddhimagga. They begin with Purification of Conduct and end with Purification by Knowledge and Vision, which itself unfolds in four stages of paths and fruits leading to liberation.
The five paths are a scheme developed by the Sarvastivada Vaibhashika school and later adapted by Mahayana traditions. They are the path of accumulation, the path of preparation, the path of seeing, the path of cultivation, and the path of no more learning, as described in Vasubandhu's Abhidharmakosa.
The ten bhumis are the ten bodhisattva grounds described in the Avatamsaka Sutra. They range from the Very Joyous, in which one realizes a partial aspect of the truth, to the Cloud of Doctrine, in which one benefits all sentient beings with the Dharma as a cloud sends rain impartially on all things.
The debate was formalized in the 8th century when Shenhui, a student of Huineng, made it a contest at the Chinese imperial court. Guifeng Zongmi and the 12th-century Korean master Chinul later argued that insight into one's true nature is sudden, but must be followed by gradual cultivation to attain full Buddhahood.