A text that gets adopted by more than one tradition raises an honest question: whose text is it actually? Mohe Zhiguan is studied today by both Chinese Tiantai Buddhism and Japanese Tendai Buddhism, and Tendai reveres Zhiyi as its own spiritual ancestor every bit as sincerely as Tiantai does. So it is worth being precise about where the text actually comes from and why that history still matters.
Zhiyi delivered these lectures in China in 594, and his disciple Guanding compiled them into their finished written form entirely within Zhiyi's and Guanding's own Chinese Tiantai context. The Japanese monk Saicho did not found Tendai until the year 806, more than two centuries later, and he did so specifically after traveling to China and studying at Zhiyi's own mountain, bringing the system he learned there back to Japan and adapting it into a new, distinctly Japanese school. Tendai, in other words, inherited Mohe Zhiguan. It did not produce it.
That distinction is not a technicality. A text created by a tradition's own founder, to articulate that tradition's own core practice, before any daughter tradition existed to inherit it, sits in a different category than scripture or canon that predates and is simply shared by multiple schools, the way the Lotus Sutra itself is shared across nearly all of Mahayana Buddhism. Tendai's later, genuine devotion to Mohe Zhiguan does not make the text any less what it was at the moment of its own creation: Tiantai's own founder, teaching Tiantai's own method, in Tiantai's own China, two hundred years before Tendai existed to receive it.