Hello, I see you posted the same thing on the discord, but I'll also reply here so anybody with the same question can benefit from it.
The grey lines that form the mesh triangles are never visible outside of edit mesh mode.
A way to see these internal lines is to manually create edges, shown as orange lines:
Mesh attachments - Spine User Guide: Edges
If you look at the example from the other user that you posted, you are in fact only seeing vertices and orange edges, but not the grey ones that Spine automatically creates.
The checkbox Wireframe makes a mesh visible as if selected, so with the vertices showing, even if it's not selected. This is very useful when using a mesh as a reference for creating another mesh or placing bones. You see no difference because you have the mesh already selected.
Mesh attachments - Spine User Guide: Wireframe
You mentioned you wanted to improve the structure of your mesh on the discord, the 3 articles linked here may help you with the task:
Mesh attachments - Spine User Guide: Vertex placement
