The priest today at Low Mass offered the English text of this wonderful prayer as a meditation for us on the feast of the Exaltation of the Cross.
This is one day, by the way, why I like in some ways the less accurate translation. The name "Triumph of the Cross" suggests the victory of sin and death achieved by Our Lord through his death on the cross.
On the other hand, "Exaltation of the Holy Cross" is beautiful. It gives the sense that the Cross is holy, and in Vexilla Regis even says that it is an altar. We honor the instrument of His Passion, for it was through a death that in one sense was unnecessary. God did not have to condescend to us. In His omnipotence He could have redeemed us in another way.
But no, Our Lord Jesus Christ, the eternal Word became flesh, and dwelt among us, so that the whole of our lives might be sanctified from birth til death.
It is interesting that the ancient pious legend tells of Adam taking a piece of wood from the tree in the Garden of Eden, which was planted and later cut down to be used as the wood of the Cross. Our Lord thus brought to perfection the Garden. Creation was not perfected in one instant but is rather brought to perfection gradually, and the old creation is fully surpassed with the new creation in Christ Jesus brought about fully at the Second Coming. We see, I think, a taste of this in the legend on the origins of the wood of the Cross.