AirMyne is trying to remove CO2 from the atmosphere and sequester it attached to rocks underground. Pumping surface water down sequesters surface water down deep that recently collected CO2 from the atmosphere, and exposes incrementally less-saturated surface water to collect more.
The important difference is not how the CO2 is collected, or where it ends up. Ultimately, what matters is how much mass of CO2 is collected per unit cost, and how much CO2 is collected in total. I.e., how profitable is it, and how much difference does it really make? We need methods that can move CO2 out of the atmosphere by, ultimately, billions of tons per week.