It happened before the East-West split.
Late-delayed baptism seems to have started (at the latest) in the 4th century, coexisted with some kind of "credo" baptism and some kind of infant baptism, and was pretty much gone by the 5th century. It was not so much "false", but that Augustine was a commanding intellect who argued well for doctrines of the church such as infant baptism.
He was convinced that his baptism was a means of grace that helped to regenerate his soul and free him from his sexual sin.
He used infant baptism as evidence for the sinfulness of infants. Thus, it was a weapon in his fight against the Pelagians.
This idea was not held by all the Church Fathers before Augustine. He cemented the power base of that doctrine, so to speak.
According to this exposition at OCA.org (https://oca.org/orthodoxy/the-orthodox-faith/worship/the-sacraments/baptism), one facet of baptism is declaring yourself "with" or part of the Church. Thus I infer that if one calls oneself a Christian, one would be baptized as a declaration.
Also, it looks like it is at baptism that one receives the gift of the Holy Spirit.
Those two points alone seem enough to command baptism as early as possible.
It reads to me as though the Orthodox church has ALWAYS believed this. Why Basil and Gregory postponed baptism may simply be bad theology on their part.
EDIT No. 1
It seems as though Basil preached against late baptism!
EDIT No. 2
EVEN THOUGH THIS IS NOT AN EASTERN ORTHODOX ARTICLE, it looks to be informative, if its quotes from the Church Fathers are correct. Caveat emptor.
This link seems to indicate that people were not postponing baptism, into the 3rd century.
However, the second part of the post suggests there were multiple views on baptism from the 3rd century -- mainly including clear indications of infant baptism. https://sharperiron.org/article/baptism-history-part-2 But the "postponement" of baptism didn't refer to old age, but to maturity:
“Jerome was born in 347…[near] Dalmatia…Jerome’s parents were Christians, who took care that he had been, as a baby, ‘nourished on Catholic milk,’ he was not baptized as a child…but as a young man…In those days, baptism was postponed until maturity…Augustine and Jerome’s friends, Rufinus and Heliodorus, are parallel cases.”15 Again the pattern was to delay baptism until maturity.
The article pins the act of delaying one's baptism unusually long times is probably from the 4th century, and lasting into the 5th century:
Many in the third and fourth centuries viewed baptism as a one-time total washing that could never be repeated. Thus, backsliders have no hope of cleansing.
If the article is correct, then it looks to me like there were at least three competing views on baptism starting in the 4th century until the church settled on infant baptism in the 5th century.
By the 5th century, the Catholic Church was solidifying its teaching of infant baptism. Augustine had influence on this, which will be discussed in Part 3.
(Part 3: https://sharperiron.org/article/baptism-history-part-3)