The difference between exercising, working out, and training
Exercising is just that, getting one's heart rate up without any goal in mind, or a very vague goal (such as "I want to lose some weight").
Working out is exercising with a vague goal in mind (I want to get stronger, I want to lose 20 pounds). It is generally done from a pre-planned set of exercises.
Training is working out with a specific (SMART) goal (I want to increase my bench press 5 rep max by 50 pounds in 12 weeks, or, I want to lose 20 pounds of body fat in 9 weeks). Training is specific to the goal, it is programmed with progressions and regressions. It also has a focus on form (how one does each exercise).
WE train people. We help establish SMART goals, and we get people to those goals.