The document model treatment you describe is talked about some in the Shapiro et al. paper as a "continuous sequence", and is roughly what was used by Logoot and Treedoc. The latter is explored more thoroughly here: http://arxiv.org/abs/0907.0929.
I was only talking about network APIs in the original piece. The "set of names" bit was there to establish the lineage between "classic" programming language/library APIs and those that touch the network.
APIs themselves do exactly nothing. It is the computational service on the other side of an API that does something. This conflation is exactly the sort of thing that is allowed and encouraged by the construction of APIs as "just another function you call in your runtime".
I find the Facebook examples you offer very curious. APIs have no inherent model for authentication and authorization, and the same goes for CRDTs. So, why do you think that verifying authorization over a set of operations or set of modifications to some state is any different than verifying authorization on N operations attempted via N API endpoints? I'll certainly grant that the latter comes with a body of current programming practice and infrastructure, but that hardly an endorsement of its relative quality or suitability for the job-to-be-done.
My preferred characterization is that the Facebook API would be replaced with a data model. The original piece already hints at a number of advantages to such an architecture, and omits many others that I'll talk about at a later date.