The question of Persian religion and imperialism is extremely complicated, and i would be very wary of "Holy War" as a rubric under which to discuss this. I will however clarify some of the cosmology involved and suggest some further reading.

What you are observing about lies and truth is a basic duality of Mazdean thought, the cosmic division between Aša, "Truth" and Druj, "Lie". These however are not specifically about 'truth and lies" but rather general principles encompassing all that is positive physically and morally in creation and all that is negative or evil. Aša encompasses justice, honesty, upright behavior, ritual purity, and good government as well as a physically hospitable clime, for instance and Druj can encompass dishonesty, bad government, the violation of contract, hatred, deceptiveness(indeed, the material existence of Ahriman and his agents, the embodiment of druj, is a sort of deception itself). I wouldn't however characterize Persian imperialism as a extended campaign of "holy war to stamp out The Lie" so much I would suggest that the Old Persian inscriptions present a sort of intellectual framework in which the king's enemies become enemies of Ahura Mazda and hence captives of The Lie. I should be extremely clear, however, that not all scholars of Achaemenid religion focus primarily on its imperial aspects and "attacking The Lie" certainly did not extend to stamping out non-Mazdean cults because the Achaemenid state maintained varying degrees of close ties with local religions under Darius and Xerxes. But when we look at Bisotun, there's certainly a degree to which Darius is framed as responsible for restoring order to his empire by extirpating The Lie from his land and the rival kings are framed as cosmic malefactors. Hopefully this helps clarify the rhetoric of Bisotun; feel free to ask followup questions.

Suggested reading on Persian religion: The foremost scholar presently active and working on the imperial dimension of Achaemenid religion is Bruce Lincoln, who has written a book on the subject called Religion, Empire, and Torture:The Case of Achaemenid Persia. It's a very accessible book, but quite strongly polemical about American imperialism and doesn't delve into the source-material as deeply as Lincoln's other book on the subject Happiness for Mankind(which however, is far, far denser). For a different perspective on Achaemenian religion, which emphasizes its roots in indigenous pre-Persian Elamite tradition, you may wish to consult Wouter Henkelman's The Other Gods Who Are; bear in mind however that it is an extremely dense and technical book with a heavy emphasis on close reading of a series of economic documents excavated at Persepolis.