Illustrator will occasionally throw in Helper points on a mesh. These helper points are not immediately visible on the mesh. Their presence is often indicated by a sharp corner or direction change in a mesh path which doesn't seem to adhere to the bezier handles and points you can see.

Your image would seem to have a couple of these helper points.

To remove the Helper points, switch to the Subtract Anchor Tool (Just tap the - [minus] key on the keyboard).

As soon as you switch tools, you will then actually see the helper points, and can subsequently click them with the Subtract Anchor Tool to remove them. Adobe states the Helper points are smaller than actual mesh points -- that may be true, but to my eyes they aren't really much smaller.

When building a mesh on a complex shape, you often need to switch back and forth to the Subtract Anchor tool to kill the auto-generated Helper points. Basically, every time you add a new mesh path you need to check for helpers.

And it's also helpful to realize that you can place smaller meshes on top of solid shapes to create dimension. You do not ever need to build 1 mesh for 1 object.

Also be aware, you can slide a mesh point along it's path by holding down the Shift key and then click-dragging the point with the Mesh Tool. (You can't slide or edit Helper points. You can only move or remove them)

Ultimately trying to build 2 meshes which are identical, on 2 different systems, may a lesson in frustration unless you start with the exact same shape containing the exact same number of anchors in the exact same positions. And then you know exactly which order the mesh points where added and click in the exact same position. Meshes are a bit more forgiving than that. As long as you get relatively close, you should be able to create the appearance you want.