It's just semantics to me. Exercise connotes general health, but training connotes a specific goal or event.

I race, so for most of the year I'm on a plan for specific races or a series of races. There are specific targets I'm trying to hit, to cause specific adaptations, and if I follow the plan, I get stronger and faster (or at least I don't get any slower).

So, if I were at a cocktail party and someone came up and asked me what I like to do for exercise, I'd say "I ride my bike." But if I were talking to another rider, I'd say I'm training for a particular race, or cyclocross season, or whatever.