Is your natural carbon sink continually growing its biomass, each year containing more biomass than the year before? Then it's an actual carbon sink. If it's at equilibrium it's a carbon store.

That's also important! Don't cut it down, for the love of our biome, but don't pretend like you can keep burning old trees you pumped out of the ground just because you have a pile of fresh trees just standing around.

If we want to fix our carbon balance we need to first of all stop digging and pumping more carbon out of the ground, because eventually that will all end up in our air. But we also need to start putting it back in the ground where it came from, or putting it somewhere else where it won't go into the atmosphere for a long time, preferably for centuries or even millennia.