Is there a standardized way to define weather icons based on quantifiable data?
The icons that are used are really just a visual way of describing the phrasing that is used in the forecast.  You can see an example of one set of icons offered by Weather Underground here.  And they have many different icon sets that you could choose from.  The phrases themselves are linked to specific ranges of probability of precipitation, cloud cover, or other factor.  This site shows some nice tables that give some of those ranges (sample shown below):