How do Catholics respond to the claims that the cross has pagan origins?

According to An Expository Dictionary of New Testament Words (London, 1962), W. E. Vine, p. 256. : >The shape of the [two-beamed cross] had its origin in ancient Chaldea, and was used as the symbol of the god Tammuz (being in the shape of the mystic Tau, the initial of his name) in that country and in adjacent lands, including Egypt. By the middle of the 3rd cent. A.D. the churches had either departed from, or had travestied, certain doctrines of the Christian faith. In order to increase the prestige of the apostate ecclesiastical system pagans were received into the churches apart from regeneration by faith, and were permitted largely to retain their pagan signs and symbols. Hence the Tau or T, in its most frequent form, with the cross-piece lowered, was adopted to stand for the cross of Christ.” There are many other references that agree that prior to 3rd or 4th century only non Christian groups ever used the symbol of the cross in their worship. There is also the matter of whether or not Jesus was put to death on a cross or a single upright stake as the Ancient Greek words stauros indicates. Thus I am asking: How does the Catholic Church explain the cross becoming acceptable to use and venerate as a symbol of Christianity?