Would the word 'conquer' serve as translation as well as the word triumph, do you suppose?
Trying to distinguish them in my head when I'm praying them, I think Of "vincit" as Him breaking in, smashing down the walls; "regnat" as Him sitting on His throne and putting everything in order all around him; training his cadre of ministers, legislating; "imperat" as ordering His law to be proclaimed throughout the world, His forces arming, and justice done.