The green tick/red cross only indicates if an object is enabled or not. You can see it with deformers, fields, primitives (which are generators anyway), other generators such as the cloner or extrude, etc. The tick does NOT show that the object is a generator. You only see it with objects whose actions can be enabled or disabled, so you won't see it with polygon objects or things such as the floor object.
To get the status of the green tick for objects which have it, use GetDeformMode().
(Edit) I've never used it, but if you call obj.GetInfo() and test for the flag OBJECT_HASDEFORMMODE, if that flag is present then it seems that the object does have the green tick. This could be useful when looking at objects which aren't generators, etc. but still have that tick.
Steve