As a synagogue congregation grew it would need a presiding ruler of the synagogue congregation (which is how the Anglican-Episcopal system, and the post-Vatican 2 RC system, actually works) and/or a resident theologically trained rabbi. Interestingly enough Professor Greg Bloomquist functions as the resident rabbi in the St. Mark's congregation of the one Church in Ottawa.
But what is the relationship between -loutron- and the Spirit? Here are four models that I encountered as a theological student.
1) A -loutron- washing in the name of the Trinity is instrumental in saving us by imparting the Spirit with the result that the person is thereby born again.. Without it the unbaptized go to hell or limbo. This is why at King's College Hospital, London, my wife who was a Baptist, was required to baptize babies in imminent danger of dying.
2) The -loutron- by immersion in water is a witness given to their faith by those who are already saved because they have believed and so been regenerated by the Spirit. This is why those who were given a -loutron- as babies were not baptized at all, and therefore need believers' baptism.
3) Regeneration is a -loutron- of the Spirit by the Word, and the outward ritual of water baptism is not needed for salvation.
4) The -loutron- of Christian baptism is a sign of God's New Covenant with us, just as a circumcision was a sign of the Old Covenant. That makes it appropriate for children, as was circumcision, with a view to subsequent faith. Life in a community of the Spirit and the appropriate fruit of the Spirit is the evidence that the person is indeed saved from hell and is among the elect for heaven.
There are many variations of these four models, and no doubt others I haven't encountered. For most Christians it makes no difference which of these models is given by way of explanation in their denomination. If they ask, they will be told that one of these models can be "proved' by the exegesis of three or four New Testament texts. But it seems to me it is not lexical exegesis that settles the question.
As a missionary for eleven years in India, and then in parish work in Canada, I was not comfortable with any of these four models. Eventually it was a text about Jesus' practice that suggested another model. -Iysous pleionas mathytas poiei kai baptizei y Ioannys- "Jesus is making more disciples and baptizing them than John." This suggested that both John the Baptist and Jesus used a -loutron- with water to enrol disciples to begin learning with them.
In India a guru would mark a person's forehead or give a flower as a sign that he was enrolled as -chela- a disciple. The person would remain with the guru for a while and keep coming back to learn more. So in my Anglican parish work I decided to try using -loutron- pouring with water in the name of the Trinity as the means of enrolling those who wanted to learn from Jesus by the Holy Spirit. This fitted the Great Commission nicely - mathyteusate panta ta ethny baptizontes autous eis to onoma tou patros kai tou uiou kaitou agiou pneumatos didaskontes autous trein panta osa eneteilamyn umin- "Go make disciples among all nations by baptizing people in the name of the Trinity and then teaching them all that Jesus had taught about God the Father, God the Son, and God the Holy Spirit" (Matthew 28:19).
That gave me a fifth interpretative model for Titus 3:5 and for my work as a parish priest in particular:
5) Jesus used -loutron- to enrol his disciples to begin learning in the school of the Holy Spirit. And in the great commission he commanded this to be done among all nations. After his death and resurrection his disciples went on enrolling people by loutron' to be taught in -ekklesia- a school of the Holy Spirit. With this model it is appropriate to baptize babies since the Holy Spirit has already begun teaching them by the love and prayers of their parents. Salvation then becomes a matter of heart direction like Abraham's faith (Romans 4), which may precede or follow teaching by the Spirit.
I am very comfortable with model 5. I therefore translate Titus 3:5
"he saved us, not by works of -dikaiosune- righteousness which we
have achieved but according to his mercy and grace by washing us -dia
loutrou palingenesias- to give us a new birth into his church where
we can be renewed by the -anakainoseos- renewing of the Holy Spirit."
There is no proof of that by the exegesis of a few texts, but it seems
to fit what God had in mind in Genesis 1:27, and the invitation
in Revelation 22:20.