Creating Quads from points

I am creating a mesh from points using Delaunay Triangulation method. After that, I was hoping to quadrangulate as much triangles to quads as possible for analysis. 

I have tried using the tools under Mesh -> utilities -> quadrangulate but the results are not as expected. Some very obvious (I think) triangles should have been quadrangulated. (See photo 1). I have played with the options in quadrangulate, the angle and ratio, but seems there's nothing difference with whatever number I throw in...

The second method I have tried is using the weaverbird constant quad split. The results are very good, but it also creates some points in the middle of one the edges of a mesh (Photo 2). Further exploring found that there was mesh connecting the three points (Photo 3). This causes error in the analysis software. 

Any idea how to resolve it? Other suggestions are welcomed. All I want is a mesh with less number of triangles. 

Thanks!