Take my post with a big pinch of salt, I may be entirely wrong here (and welcome anyone to correct me entirely!)
First thing to note: in the paper they are using quadratic splines, not cubic ones. As a result the numbers of their control points and knots won't match the configuration you have above.
I 'think' this is their equivalent of their cox-de-boor (ish) for a triangle. I'm guessing the 3 terms on the right [B(u|K\v0), B(u|K\v1), and B(u|K\v2)] are simply the standard cox-de-boor algorithms?
B(u|K) = a0B(u|K\v0) + a1B(u|K\v1) + a2B(u|K\v2)
If a0, a1, and a2 are just the barycentric coordinates of the point being tessellated, then I am assuming the 3 basis functions would look something like this:
a0 = u
a1 = v
a2 = 1 - a0 - a1
That makes sense to me anyway, since the blend weights in that configuration should still sum to one?
Then there is a bit about how the last term in the above equation can either be 1 if the 'u' point are in the triangle formed by the three vertex knots and 0 if it is not.
Which is why it sounds a lot like the cox-de-boor (which always terminates the recursion by returning 0 or 1 depending on whether the point affects a given 'u' value).
I think that talk about knot configurations is possibly a red herring. I don't think (buy may be wrong) that it has anything to do with the algorithm, just the knots chosen in the respective directions. Mind you, I might well have accidentally just defined a Tensor-Product spline by accident (I can understand NURBS just fine, but those papers make my head hurt!)