The lines ‘on the surface’ won’t be related at all with the surface an its mesh, they will generate its own line elements.
the lines must be boundary of the surface
You can join the ‘internal line’ with the boundary, creating extra auxiliary lines, and then can split in two surfaces
and probably, to simulate a crack you don’t need a single line shared by both surfaces, because the domain will be continuous, sharing the nodes.
You can ‘uncollapse the surfaces’ to duplicate the lines and points. (and maybe you simulation require a kind of link between the pair of lines to consider it as a contact)
or with a single shared line can assign Mesh->Mesh criteria->Duplicate->Lines,
to create mesh nodes duplicated for each side.