Yes, as that link says, it’s only organic if organic acid is used. Citric acid. Acetic acid (vinegar).
There could be other ways. I think if the water was warmed up and/or a vacuum pump was used to pull a vacuum on the water, then most of the carbon dioxide (CO2) would leave the water. This should cause precipitation of the relevant mineral solids like calcium carbonate and magnesium carbonate. The solids are easily removed from the softened water by decanting and/or simper (paper)-filtering. No acids needed.
A simple solar water heater could be used with a simple counterflow heat exchanger along with a solar powered vacuum pump. Then slowly fill a water storage tank when it’s sunny, or skip the tank and only irrigate (at a low flow rate) when it’s sunny.