There are a couple of very different ways of understanding notions such as "buddha nature".
Those very brief posts above only represent one of them, which tends towards the Tathagatagarbha theory side of things. Even this teaching has several forms, so a single textual citation will be too brief. But in general, it takes the Tathagatagarbha / Buddha nature idea as definitive over the other teachings, such as non-self. It may claim in some cases that non-self is applicable to certain phenomena, eg. the aggregates, but not to the Tathagatagarbha, which is subtle and difficult to perceive. However, the idea is that every living being has this buddha nature within them, a fully awakened buddha ready to be uncovered. This means that this type of buddha nature theory is only applicable to sentient beings, but not the insentient.
The other main explanation is that "buddha nature" refers to the emptiness, dependently originated nature of all phenomena. It thus makes the emptiness teachings definitive over such teachings as a true self Tathagatagarbha, etc. It considers that this buddha nature is not some thing within the heart / mind of each living being, but is merely potentiality. ie. because phenomena are empty, they can be enlightened. This notion of buddha nature as emptiness may thus be applicable to all phenomena, not just sentient beings.
Both of these two main schools of buddha nature thought have many subtle sub-schools and ideas, too.
Some schools, such as Huayan in East Asia, and mid-period Chan / Zen, will tend towards the first type as definitive. Others, such as most Madhyamaka based schools, will take the latter. They are in many ways very very different takes on the same words / terms. Often people will discuss this topic, and fail to notice the main differences. They then tend to talk past each other. It is thus worth clarifying before continuing further with such discussions.
Venerable Huifeng, while those may be important academic, historical clarifications, they do not seem to be vital distinctions for my experience. To borrow a cliche metaphor you are probably familiar with, one can use many different types of fingers to point to the moon.
You can take Buddha-nature as definitive over emptiness, emptiness as definitive over Buddha-nature, state that Buddha-nature and emptiness are the same thing, or as Theravadins do, you can even deny (or without denying can simply refuse to teach) Buddha-nature and the Mahayana understanding of emptiness. All of these things may in practice be either fruitless ideation or fruitful analysis, because what defines their fruitfulness is not their outer appearance, but their essential nature. Do you know what I mean by that? Meaning is something ascribed by mind (it could also be said to be intrinsic to reality but let's ignore that for now), which is not intrinsic to language, so the fruitfulness depends upon the terms as they are understood and used by minds
. People can use one or the other sets of language, while still engaging in mental activities which are more or less fruitful. When this happens and someone uses a certain language and practice still finds themselves engaging in fruitless mental activities, the dogmatic one says, "They don't truly understand the teaching," but in reality, whether Theravada, Zen, Tibetan, Huayan, Madhyamaka, whatever; the dogmatic one does not understand his own teaching
. By dogmatic, it means attachment to views, regarding views as "mine" and "not mine," and reflecting, "This and only this is the Truth and nothing else."
And it's not that I'm attached to my own expression either! In that regard, perhaps your distinction was
useful and we could benefit from knowing these many distinctions, if the people that read this may benefit from one or another particular expression.