Your definition makes a lot of sense to me. In pure programming languages, an object can appear when you run the code, but never before that. The pure description of the program cannot contain objects -- since objects carry state -- but when this description/code is executed/evaluated by the runtime system, a stateful object appears that you can interact with (e.g. a web server).

So, for pure programming languages, this "private state" is contained within the runtime system (out of reach for the programmer), rather than in the code itself, and the RTS applies pure functions to the state it carries, in order to reach a new program state.