I believe "sourdough" and "natural leaven" are the same. They're a collection of yeasts and bacteria
trapped from the air by a mixture of flour and water. I got mine started with a recipe from the LA Times years ago that started with pineapple juice and whole wheat flour, and I've had it going for years despite long stretches of neglect at the back of the refrigerator.
Commercial yeast is a single strain that is very efficient, but the theory is that the multiple strains in a sourdough do a slower but more complete fermentation, resulting in a more digestible and nutritious bread. Even a white flour loaf of sourdough bread is supposed to have a lower glycemic index than a white loaf made with commercial yeast.
I can vouch for the slower rise!
A lot of the "sourdough" bread we grew up with was made with commercial yeast too, using the starter more for flavor than function. Those breads are quite sour. I think changing the name to "natural leaven" is an attempt to get away from that image, and to make it clear there is no commercial yeast in the bread. I've found that using a small amount of starter at first and nursing it along gives a really flavorful result that is not very sour at all. In France, they call it "pain levain." They don't keep a starter going; instead they keep a piece of dough from each day's batch to begin building the dough for the next day.
I'm not a purist. I use commercial yeast for baguettes, pizza dough and focaccia, but I'm trying to do more sourdough and more whole grains for everyday bread.
https://cooking-ez.com/four/recipe-natural-leaven.html