Carbon sink and carbon storage
According to Wikipedia:


  A carbon sink is a natural or artificial reservoir that accumulates and stores some carbon-containing chemical compound for an indefinite period.


The key point here is the "indefinite period" part. For example, although my body stores some kg of carbon, I can't qualify as a carbon sink because most of it is likely to be released again in some decades.

On the other hand, soils store large amounts of carbon, and as the soils keep developing that amount tends to grow, making them an effective carbon sink. The mechanism consists mostly on accumulating organic matter, as you say, but as long the amount of that organic matter keeps growing, soil works as a carbon sink.