A particular uniqueness provided by language as a fundamental means of communication is self-awareness and the ability to mediate the processes of communication. This adds a dimension that is of a different order. We'll explore what that might be at some later time. I have not studied semiotics and may be getting into trouble with my response to this. At that level, I'd agree with your comments that a distinction may not be necessary or useful.
Language provides a self-organising ability that is beyond sign using.
Language is self organising for itself. It also organises the users of the language. The users of the language are also its continuing creators. The interplay between all of these allows us to create a grammar or syntax with which we can influence the patterns of organisation of our own medium of communication - maybe even existence because we live largely in a socially created world and most of our interactions are with or determined by that world. An appropriate theory of organisation for the third level - being developed by complex adaptive systems research, autopoetic research and others provides a great deal of useful thinking for the fourth level task but doesn't completely fill the need.