There’s no confusion about how the model is constructed in this one because of the correctly provided full interface screenshot in the OP. It’s just the term that bugs me.
The problem with the term “mesh” is that people use it to describe a model with one object, multiple objects, connected and unconnected geometry. No one asking questions refers to a mesh as the object data datablock, not knowingly at least.
With modeling questions it’s important to know if the user is talking about one or multiple objects, and if the object in question has one connected, or multiple unconnected mesh parts. Object and mesh are both singular, and talking about either datablock doesn’t take into account how the mesh elements are linked. Object and object data datablocks are the same in that regard, they both contain the same data, one is just higher up the hierarchy.
Talking about objects, which are manipulated in object mode, it’s clear what the user refers to. Also talking about meshes, as multiple mesh parts, it’s clearer that there are multiple unconnected parts which one edited object consists of.
That’s the long way of saying: objects in object mode, meshes in edit mode. Plural when there are multiple, and it’s much clearer, at least to me.