Cannot Add Edge Loop to Geometry created using Cube Grid

I want to create an edge loop to add an edge around the mirror point of this mesh I made using the CubeGrid feature of Modelling Mode:

This model was created by Pulling a large block, and then pushing down on the innermost top faces to create a sort of bucket shape.
It has optimal topology, using two operations to create (Comprised of 14 Polgyons/28 Tris)

I want to add the edge loop running down the mirror line of the mesh, as I’d expect to be able to do using a standard modelling package (Faces I want to add edge to are highlighted below:

But the places I can put edges don’t seem to make sense, given where Tri editing mode believes the triangles are. below:

Strangely, if I use Poly editing mode to create the inset and interior from the initial block, it works as I’d hope. But I have found it more convenient to use the Pull and Push operations in CubeGrid.

Is there something I can do to make this work? Is it a workflow problem and I just need to do things differently? Or is it a quirk of the system and there’s something I need to do to my mesh to make poly mode respect the Quads the meshs hould have?

Thanks in advance!

The TriModel tools edit a mesh using triangles. It sounds like you want to add an edge loop to a quad not a triangle. You will need to use the Edge Loop tool in PolyEd vs the TriEd tool. The second part of the answer is how PolyGroups are generated on your mesh.

PolyModel tools edit a mesh based on PolyGroups, which are a grouping of triangles (& can be used to mimic quad based modeling). Grouping the triangle faces should give you the results you see after using inset in PolyEd.

You can group your triangles using the Grppnt or GrpGen tool under the Attributes category. Then use the PolyModel tools. Using TriModel will ignore PolyGroups aka your quads.

See Understanding PolyGroups to learn more. The FAQ section touches on edge loops on triangles.

Video for reference

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My issue is worse, sorry for old post but its close enough to valid for my issue.
I made a standard cube, added edge loops and they do as expected show up on outside and inside of cube, but on a bit more complex mesh, they aren’t showing on inside at all as I add them, how to fix ?