While the AC controller can be powered by 12VDC power source without any problem, it cannot switch valves off, due to the way triacs (i.e. AC transistors) work (there needs to be a zero crossing in the current in order to switch triac off). The DC-powered controller addresses this issue by using MOSFETs to switch valves and not triacs.
If you have experience with electronic circuits, you can modify the AC controller by replacing the triacs with NPN transistors or N-channel MOSFETs. Otherwise, using 24VAC relay also works (though it will likely involve a lot of wiring).