The first paper I am aware of in the literature that has addresses analysis of non-midpoint splitting is:
Pav, S.E.; Walkington, N.J., Delaunay refinement by corner lopping, Proceedings of the 14th International Meshing Roundtable (2005) pdf.
The motivation of that paper was a little different: they authors were analyzing refinement of curved inputs. But the analysis applies to non-midpoint splitting of edge splitting: a split point of a curve is similar in that it is not at the center of the circle with diameter between the segment or curve's end points. The take away from the analysis is that non-midpoint split points can be used but at the expense of a reduction of the guaranteed quality that can be ensured in the algorithm output. In the paper, this is seen in the appearance of $\mu$ and $\eta$ in Corollary 3.
The paper of Foteinos, Chernikov, and Chrisochoides (here) also analyzes non-midpoint splitting although with a different dependence between the parameters: the minimum angle in the output is fixed and the algorithm reverts to midpoint splitting whenever there isn't enough space to permit a midpoint split and guarantee mesh quality.
Are there split points that are better than the a-priori selected
There isn't any good evidence of non-midpoint splitting benefiting a particular application. For triangle circumcenters, there are two common motivations for using non-circumcenters,
Intentionally create triangles slightly above the quality threshold. The goal is to spread points out as much as possible without triggering further refinement allowing the algorithm to terminate with as few vertices as possible. This was the original motivation in Üngör's paper and Triangle does this.
Deliberately create triangles with angles hopefully well above the thresholds of guaranteed termination. Acute does this and on many cases is fortunate enough to create meshes that outperform theoretical expectations.
Figure 7 in this paper shows how the type of off-center selection can be used to manipulate the distribution of triangles created in the final algorithm. The results aren't that surprising: if you go out of your way to create a during type of triangle in the algorithm, then you create lots of that type and can see them over-represented in the distribution.
Non-midpoint edge splitting hasn't been particularly effective an addressing these needs. While circumcenter splitting is done in a way that to avoid creating subsequent poor quality triangles (when filling a empty space in the domain), the midpoint is generally the best position with the local information available. If the algorithm could look ahead 2, 3, 4, ... steps and see what subsequent refinement would be made to the split segment, maybe there could be some better prediction on how the segment should be refined. This paper suggests that if you know the local feature size (i.e., how long the segments needs to be in the final mesh) up front, then you can make better decisions on where to place split points and provide better guarantees on the final output. But for standard Delaunay refinement, no one has found information available at the time of an edge split that justifies a non-midpoint split.