probably mostly pedantry, but a lot of things can naturally store carbon, so maybe trapping carbon is specifically the unnatural capture of it?

In the first case the carbon dioxide is already concentrated, and in the second it has to be extracted by processing (at least) 2500 tons of air for each ton of carbon dioxide obtained. There are easier cases for carbon capture, when CO2 can actually be captured at the point of release (steel and cement plants, landfills) but atmospheric extraction is hard. Of course, plants can and do process lots of air (by it blowing over the leaves) but massively increasing plant growth is also hard.