What are "Active Calories"? The Apple Watch Activity app purports to measure what it calls "Active Calories" which it counts towards its "Move" goal. But Apple's definition of such calories is clearly its own, not counting (at least apparently), for example, calories from weight workouts, Pilates, or yoga or 7-minute workouts?

What is the (Apple) definition of "Active Calories"? What does one have to do to generate them?
Apple doesn't define active calories in the manual for Apple Watch, but they are a common measurement in sports physiology.

If you are in a coma, your body uses up calories from two of the three muscle types: smooth and cardiac muscle. The third type is skeletal muscle and that's what Active Calories tries to estimate using a physiological model that clearly considers average human statistics and then incorporates input data you feed Apple's Health App (height, weight, sex, heart rate) and then uses the accelerometer to pick apart your specific activity level for input to the model. 

Presumably a large amount of PHD level research went into Apple's internal algorithms and models based on who they hired to work on the fitness teams in several internal laboratories.

Since you mentioned some sports, here are my thoughts:

The watch can't know about what weights you are lifting since it doesn't have sensors for that (yet). In fact, the algorithms may mess up since it's seeing your wrist move slowly yet your heart rate goes up much more than it might expect/anticipate. Active calories will be far more accurate for people that run, walk and move without machines or weights attached to their bodies. So pilates (non-reformer), yoga, and 7 minute workouts without weights will be more accurately reflected in Active Calories than will reformer pilates and weight lifting.

Hopefully, weight machines or extra data input can arrive some time later for people to get excellent results with weights. I could see non-free weight machines interface with an app or health kit directly to enhance the recording of your workout profile and progress. Also, indirectly your heart rate will reflect your weight lifting, so without needing specific input, the active calorie measure should still be useful in an absolute measure and highly useful for comparing one workout to another as you add weights and change your metabolic response to physical stress.