reference for a special modular decomposition
Indeed this concept exists. Two vertices $u,v$ that have the same neighbors are often called twins. They are called true twins if they are also connected, and false twins otherwise. In your definition, all vertices of $X$ are true twins.

It looks like there are several other names in the literature for these concepts. See e.g. this MO question. In FPT algorithms we sometimes call the number of twin classes into which a graph can be partitioned the neighborhood diversity of the graph, see e.g. here (warning: self-citation!), and here for the related notion of twin-cover.