I think the problem is you're conflating two concepts, the shifts in supply and demand individually and the effect of a shift on the way they interact.
The individual curve shifts do not cause changes in market-clearing price/quantity - that requires both curves to interact. The supply curve and demand curve separately describe the relationships between quantity supplied/price and quantity demanded/price, but not the market-clearing quantity and price.
When the supply curve shifts to the left, it means that at any given market-clearing price, the quantity supplied will be lower. This is because whatever shifted the supply curve to the left increases marginal cost, as you state in the question. Imagine just a supply curve and a horizontal line at some price, then shift the supply curve left. The quantity supplied at that price has decreased. That holds for any horizontal line you choose.
Similarly, when the demand curve shifts to the right, it means that at any given market-clearing price, the quantity demanded will be higher. Again, imagine a horizontal line and just a demand curve, and shift the demand curve to the right. The quantity demanded has increased at the price represented by that horizontal line (and along all other horizontal lines).
Now, to know how the market-clearing price and quantity change, you need both curves. If you plot both a supply and demand curve at some starting point, then shift one of the curves, the intersection point moves along the other curve. If the supply curve shifts to the left, imagine it happening in a bunch of tiny steps. As the curve moves further and further to the left, the point where supply and demand intersect is moving slowly up the demand curve. That point is to the left and above the original point, so the market-clearing price increases and the market-clearing quantity decreases. Similarly, if you start from a supply/demand interaction and shift the demand to the right, you'll see that the intersection point moves up and to the right along the supply curve.