What is the name of our practice, the practice that descends from the lineage of Bodhidharma back to the Buddha, not a lineage of genes or bloodlines, but a lineage of no-thing, the only thing that can be transferred without loss and without gain from body/mind to body/mind?
The question is easy and difficult. Easy because naming is easy and those that have come before have already named. Difficult because with naming comes a fertile ground of delusion, where ignorance spreads like wildfire.
Case in point: try ‘Zen’. If we practice in this lineage that descends from Bodhidharma back to the Buddha, do we practice Zen Buddhism?
No. Zen is the Japanese branch of this lineage. There are questions of historical and spiritual importance that only apply to (Japanese) Zen Buddhism. One example is illustrated by this post over at Wild Fox Zen. Are Shikantaza and Koan practices compatible or mutually exclusive? Is Koan practice part of Dogen’s teachings or rejected by him? These questions (and many others concerning (Japanese) Zen Buddhism) do not arise for (Korean) Sŏn Buddhism or (Chinese) Chán Buddhism; they are not, in a sense, of spiritual importance to practitioners of those schools and need not be of historical importance either.
Yes. Since practitioners (myself included) in this loosely knit set of practices trace their lineage back to the Buddha through Bodhidharma, they must continually and sincerely confront the heart-essence of that lineage, the lineage of no-thing, the only thing that can be transferred without loss and without gain from body/mind to body/mind. If (Japanese) Zen Buddhism has developed responses to the ineffable song of the heart-essence, then the (Korean) Sŏn Buddhist or the (Chinese) Chán Buddhist ought to listen deeply. Intentionally not doing so reeks of delusion.
So the situation is complicated, to say the least. But when our discourse lumps all of this into Zen Buddhism and acknowledges the other practices by occasionally using their names, if at all, we are sowing seeds of delusion among potential practitioners and seasoned ones alike. For example, Buddhadharma says, “We are fortunate to have the support of Editorial Advisors who represent a wide variety of communities and traditions—Theravada, Zen, Pure Land, and Vajrayana.” That is not a wide variety by any means, if you take into account the diversity of responses to what drives practice. However, if Buddhadharma were genuinely sensitive to the diversity of the real world (and I have no reason to doubt the people there are sensitive in this way), they would run out of ink and room to print! Remember, the situation is complicated, to say the least.
So what is the point? Our words have karmic effects on those around us. When we speak, we shape the body/minds of those who hear. When we speak in a way that is out of tune with reality, the body/minds that listen are also out of tune with reality. When we speak in a way that is in concert with reality, the body/minds that listen are also in concert with reality. Our responsibility is to be careful with our words because that is being careful with awakening – our awakening as one body, one mind.
Recognizing the limitations of our names because of their historical existence is liberating in at least two ways. It is liberating because when we get hung up on a problem that seems of utmost importance, we can remember that it is of no importance and move towards easing our body/minds. At the same time, it is liberating because getting hung up is how we move forward with practice; knowing the how and why of our hangups illuminates the path forward. In a sense, these two ways are not two, yet as Uisang says in the Ocean Seal they are “not confused or mixed, but function separately.” 
For my own part, I will try to use Sŏn/Chán/Zen Buddhism when I hope to write about our practices in general. When talking about a particular lineage, I will use the name for that lineage. And I will fail most of the time at accomplishing what I hope to say.
For example – the astute reader will be asking about (Vietnamese) Thiền Buddhism. And the fact that we who practice Sŏn/Chán/Zen Buddhism do not all share the same lineage back to Bodhidharma, so the differences are even more extreme than we make them out to be. And that the lineages sometimes have wholes in them, gaps between generations of teachers and masters and nuns and monks. And more. Oh the doubt and the unknowing mind that permeates the heart-song of our practice!
As Juliet says of Romeo in Skakespeare’s play: “What’s in a name? That which we call a rose // by any other name would smell as sweet” (Act II, Scene II). Yes, oh Juliet, how true the words. The rose would smell as sweet were it called ‘horse’ or ‘manure’. But call a rose ‘manure’ and you might find unsuspecting lovers walking up to piles of manure suspecting that something is not so sweet about the smell.
Call it Sŏn, Chán, Zen, or Thiền and you still have the same sweet smelling practice. But be careful not to lead unsuspecting practitioners to piles of manure. And be careful to smell the practice, whatever the name, before you put your nose in it.
 This excerpt is from Uisang’s Ocean Seal of Hwaom Buddhism as translated by Steve Odin in Process Metaphysics and Hua-Yen Buddhism. You can also find the Ocean Seal in the (free and online) Collected Works of Korean Buddhism, Vol. 4, p. 103.