> I think the escape hatch is that the information conveyed is not the points on the screen, but the rule on how to generate those points. Going back to my other post this means that the city builder conveys no information about a particular city map.

Yep. The amount of information conveyed can be considered 'equivalent' to the length of the shortest algorithm used to get from one string to the other.

The field that is concerned about thinking about this problem in a disciplined manner is algorithmic information theory:

http://en.wikipedia.org/wiki/Algorithmic_information_theory

In particular, they show that the quantity is incomputable (the Komolgorov complexity).