Training is "the process by which an athlete prepares for competition by exercising, practicing, etc." (merriam-webster.com). The two key words here, especially for runners, are process and prepares.
Prepares implies a goal. In our case, a race.
Process means that we introduce structure to our running – e.g., rather than just going out and running "a few miles, a few days a week, or whatever" we run according to a plan. Typically these plans include things like long runs; speed work, such as intervals and tempo runs; recovery days; and strength or cross-training. They also include a "taper" period at the end, with mileage dipping as race day approaches. The intended effect is to turn you from a Person Who Runs Sometimes into a Laser-Focused Racing Machine Capable of Great Things.
If your training goes well, you will find yourself on the starting line of your goal race feeling like a coiled spring. (This will mean you'll probably go out too fast, even if you've told yourself over and over that you won't go out too fast. But that's another story.)
Put another way: If you're running with a goal in mind and a plan to help you reach that goal, you are training. If you are running "just to run," you aren't really training per se.
Training is running with a special purpose.