Triangulation doesn't respect points I wish to preserve

Hi @eobet
The key word here is feature preservation
The remeshing can only keep the specified points if they exist as mesh vertices in the input mesh.
If they don’t exist you’ll need to insert them. The best way to do this depends on the input (I’m not seeing a link to your file - where did you upload it?).
Here’s a script I made for inserting vertices that can be used for this

(that still assumes the points being added lie somewhere on the mesh. If you want to pull the mesh to new feature vertices which are off the surface, it would need a little addition for that)