Is it true that if you lift heavy weights you will get bigger, and if you lift lighter weights you will get cut?
Well, it is true that heavier weights and lower reps will be more effective in building size and strength. However it is not true that lighter weights get you cut. This is a myth in the fitness world that has been perpetuated for many years, and is still alive and strong today!
Getting cut or lean requires proper diet and nutrition, and while you are on a "cutting" cycle you should still train with heavy weights to maintain your muscle mass. There is a principle referred to as "use it or lose it", it basically implies that if you do not utilize your muscle mass your body will see no reason to maintain it. So, even during a cutting cycle you still want to train as heavy as possible, it's inevitable (especially if you're natural) that you will sacrifice some size while cutting. But if you train with light weights you will most likely lose twice the amount of muscle mass than if you continue to train heavy while cutting body fat.
Now, this does not mean that you should never go through "light cycles" of training, I truly believe that you should. Using short meso cycles in 4-6 week periods is extremely important if you want to continue to make progress. So, training heavy year around is not a good idea either, it will lead to over training and ultimately injury.