Exercise can be a noun or a verb. For our purposes, the noun refers to specific movements or poses, such as push-ups, planks, or running. Each of these is an exercise (n.). That part's not particularly controversial.
But as a verb, exercise (v.) can be more vague. (And we're talking fitness here…my finance friends have other definitions.) When someone exercises (v.), they perform one or more exercises (n.) but nothing else is implied. This is what most people do to try to be fit: they exercise.
The phrase "work out" (v.) in fitness is often used as a synonym for exercise. So "I will work out later" is the same as "I will exercise later." But the noun workout (n.) should have no such ambiguity. If you do a workout, it should not mean just doing one or more exercises, it should mean performing them with a plan. Not some arbitrary plan, like doing 3 sets of 10 push-ups and running a mile, but a plan built around some history (or, initially, to create some history). If the previous relevant workout was 3 sets of 9 push-ups and running a mile in 8:10, then the next workout could be doing 3 sets of 10 push-ups and running a mile in 8:05. Even if you don't hit those numbers, having them as intelligent targets takes us from exercising to working out.
These workouts should be part of a longer term plan, and that's where train (v.) (or "training" (n. or adj.)) comes into play. Normally, training is for a specific goal, such as a marathon or the start of a sports season (hence, "training camp"). But we can also consider training to be a series of workouts towards a more general goal, like increased strength or improved cat-herding ability. As long as you have a sensible plan and keep working toward it, we can call it "training." That's my story and I'm sticking with it.
Unfortunately, most people who think they're "training" are really just "exercising" because their workouts have no cohesive plan and feature goals like "get toned" or "lose 10 pounds." Many people do the same exercises every time, without increasing loads or progressing exercises (modifying or replacing to increase difficulty) to force new adaptations. People can stay healthy doing the same exercises consistently, but they won't improve without upping the ante.
On the flip side is the need for rest and recovery. Training requires time for the body to adapt to the applied stresses, so it can get stronger or faster (StrongFast!). Exercising too hard too often doesn't allow the body time to adapt. It also tends to lead to burnout: witness the annual spectacle of the transient crowds at the gym each new year.
As usual, I digress. In summary, consider these definitions:
Exercise (v.) is doing exercises (n.) without a plan for achieving desired results over time.
Work out (v.) is doing one or more exercises with a plan.
Train (v.) is doing regular workouts that progress over time towards a well-defined goal.
What do you do?