I'm just working on quadrangulate free form on MoI to use it in ParacloudGem...
My way is wrapping a squared panel on the free form with the flow v3beta tool. If I want straight lines for the quad, I export in OBJ with 180° to straighten any spline.
But the quad are not planar... :(

You can easily make planar quad in the second dome because the geometry is made with parallel scaled ellipses, so you can divide each ellipse in the same way and find parallel lines to make planar quads. But in your first shape, I cant see the geometrical way you model it. Is there any scaled curves ? There are arc or free splines ?... Send the file to see ;) [edit : oops... I haven't seen in the first post... sorry]

Object made with the first process and :