I think the Petri net is a great example, but I disagree with the conclusion that we should express something like a Petri net primarily as a diagram. The ideal model is still textual, but as a DSL that compiles both to something like DOT format and to a programming language of your choice. This gives you all the many benefits of textual formats, along with the visual economy of a diagram.
I think the conclusion is because one diagrams has many different syntactical expressions, but the theory tells us they all behave the same, so we can just pick the most efficient, most compact, whatever. ie. the diagram is the best representation, not the syntax.
so we can go: diagram -> expression -> code of your choice
from expression we can go back to diagram, but code to expression not necessarily.
the problem now is that if the picture is the leading representation, then we need to lift all the tools to the diagram realm, so comments, higher order diagrams, grouping, closures, variables, etc
