Elastic and elasticity are technical terms, so you need to understand the technical definition. "Elastic" means that the object bounces well, like a steel ball on a steel block. "Inelastic" means the object bounces poorly, like a playdoh ball on the floor. You want billiard balls to be very elastic.
Ivory balls are less elastic than cast phenolic balls. When the artistic competition changed from ivory to phenolic (about 1990?) some of the shots became much harder and some became much easier, so the program (list of shots) was changed.
Phenolic balls vary in their elasticity. I have heard that SALUC does a special test to select carom balls for the top tournaments. They bounce balls off a heavy steel block and the ones that bounce the highest are sent to the tournament.
If you want to know more, look here:
https://www.google.com/search?q=elasticity+ball