You have a few different choices when meshing for relaxation, and which is best depends on the aims.
My example used a structured quad mesh - starting from a coarse mesh with just 8 quads and subdividing. One advantage of this approach is that it makes it possible to separate the edges into warp and weft directions, for if you want more control whether the result is more funnel or cone shaped (like shown here). It can often also be a good starting point for designing the cutting pattern of fabric strips for fabrication.
As these coarse quad meshes have long straight edges, they won’t match given curved boundaries before subdivision and relaxation (which can make them a little more complex to set up, since you can’t simply anchor the naked points), but they get pulled to the target curves when running the solver as shown in my file.
If you model the pre-relaxed geometry as NURBS then mesh it using the standard mesher, like in your example, you can meet any boundary curves easily, but the mesh will usually be quite uneven, as it uses smaller faces near curve details. This unevenness can sometimes lead to some bulges and dips in the result, since the tension won’t be evenly distributed.
Another possibility is to model as NURBS, mesh, then TriRemesh. This will give near equal sized and equilateral triangles, so will give a smoother result when relaxing.
Going further than this, if the input form is far from the relaxed form, the triangles can still change size and shape a lot during the relaxation, and in this case you might want to use the LiveSoap component, which will remesh as it relaxes to keep it even.
With any triangular remeshing you don’t have the convenient directionality of a structured quad mesh though.