You won't find word "agape" in the Bible either (except in Greek). Triple immersion was a Trinitarian formula from the very beginnings of the Church, taking the Great Commission literally.
"By three immersions, therefore, and by three invocations we administer the important ceremony of baptism...and that the souls of the baptized may be purified by divine knowledge." (Basil, Bishop of Caesarea )
"You were conducted to a bath just as Christ was carried to the grave and were thrice immersed to signify the three days of His burial." (Clement of Alexandria)
"The true doctrine of our holy mother, the catholic church has always, my brethren, been with us, and does yet abide with us, and especially the article of baptism and the three immersions." (Monulus at the Council of Carthage)
"You were led to the holy pool of divine baptism as Christ was carried from the cross to the sepulcher and each of you were asked whether you believed and made that saving confession and descended three times into the water and ascended again and that water of baptism was a grave to you." (Cyril of Jerusalem)
"Christ delivered to his disciples one baptism in three immersions." (Chrysostom, Patriarch of Constantinople)
"Concerning baptism, baptize in this manner: Having said all these things beforehand, baptize in the name of the Father and of the Son and of the Holy Spirit in living water [that is, in running water, as in a river]. If there is no living water, baptize in other water; and, if you are not able to use cold water, use warm. If you have neither, pour water three times upon the head in the name of the Father, Son, and Holy Spirit." (Didache, Chapter 7, c. 70 AD)
[I]mmerse three times "thrice immersed" (Tertullian, The crown or De Corona, ch 3-4)
"[D]ipping the head three times in the layer" (Jerome, Dialogue Against the Luciferians, 8)
The Eastern Orthodox Church, unlike western churches, does everything in the name of Holy Trinity. It interprets the Great Commission to read "in the name of the Father, Son and Holy Ghost" as baptizing in the name of all three God-persons individually.
Catholic.com says that:
Although Latin-rite Catholics are usually baptized by infusion (pouring), they know that immersion (dunking) and sprinkling are also valid ways to baptize. Fundamentalists, however, regard only baptism by immersion as true baptism, concluding that most Catholics are not validly baptized at all.
Although the New Testament contains no explicit instructions on how physically to administer the water of baptism, Fundamentalists argue that the Greek word baptizo found in the New Testament means “to immerse.” They also maintain that only immersion reflects the symbolic significance of being “buried” and “raised” with Christ (see Romans 6:3-4).
It is true that baptizo often means immersion. For example, the Greek version of the Old Testament tells us that Naaman, at Elisha’s direction, “went down and dipped himself [the Greek word here is baptizo] seven times in the Jordan” (2 Kgs. 5:14, Septuagint, emphasis added).
But immersion is not the only meaning of baptizo. Sometimes it just means washing up. Thus Luke 11:38 reports that, when Jesus ate at a Pharisee’s house, “[t]he Pharisee was astonished to see that he did not first wash [baptizo] before dinner.” They did not practice immersion before dinner, but, according to Mark, the Pharisees “do not eat unless they wash [nipto] their hands, observing the tradition of the elders; and when they come from the market place, they do not eat unless they wash themselves [baptizo]” (Mark 7:3–4a, emphasis added). So baptizo can mean cleansing or ritual washing as well as immersion.
A similar range of meanings can be seen when baptizo is used metaphorically. Sometimes a figurative “baptism” is a sort of “immersion”; but not always. For example, speaking of his future suffering and death, Jesus said, “I have a baptism [baptisma] to be baptized [baptizo] with; and how I am constrained until it is accomplished!” (Luke 12:50) This might suggest that Christ would be “immersed” in suffering. On the other hand, consider the case of being “baptized with the Holy Spirit.”
In Acts 1:4–5 Jesus charged his disciples “not to depart from Jerusalem, but to wait for the promise of the Father, which, he said, ‘you heard from me, for John baptized with water, but before many days you shall be baptized with the Holy Spirit.’” Did this mean they would be “immersed” in the Spirit? No: three times Acts 2 states that the Holy Spirit was poured out on them when Pentecost came (2:17, 18, 33, emphasis added). Later Peter referred to the Spirit falling upon them, and also on others after Pentecost, explicitly identifying these events with the promise of being “baptized with the Holy Spirit” (Acts 11:15–17). These passages demonstrate that the meaning of baptizo is broad enough to include “pouring.”
It further goes on to say that much of the earliest Christian art shows either pouring or sprinkling, or if standing upright in a river, having water poured from a cup or jug over the head. An early baptismal font found in Nazareth is obviously far too small for immersion.
I’m not aware of anything we Romish do that is not in the Name of Father, and of the Son, and of the Holy Spirit. I thought that we are fully in agreement on that.
Agreed, the Western church has been guilty of a non-Trinitarian approach to things at times and in places. Each of our churches has been imperfect in some regard or another, and even when we honour the Trinity with our mouths, we sometimes do not honour Him fully with our hearts. The work of the Holy Spirit in particular come to mind.
The triple baptism though, as your texts clearly show, are a human tradition. Scripturally, we are not told that Christ/God/Spirit requires it.
“You were conducted to a bath just as Christ was carried to the grave and were thrice immersed to signify the three days of His burial.” (Clement of Alexandria)
Note the word ‘signify’. We should be careful to separate those things which are tradition (even beneficial tradition), and those things which are for salvation.
The tradition is a lovely one, and many ‘Western’ churches use it as well... it is not however a requirement for salvation.