Hi Magdalen,

The living water you referr to is Jesus. Jesus used this verse:

John 4:10
Jesus answered and said unto her, If thou knewest the gift of God, and who it is that saith to thee, Give me to drink; thou wouldest have asked of him, and he would have given thee living water.

to point out to the woman that He was the one spoken of by the prophets:

Jeremiah{2:13} For my people have committed two evils; they have forsaken me the fountain of living waters, and hewed them out cisterns, broken cisterns, that can hold no water.

The word used for born(gennao), in Greek (in those verses) has nthing to do with inititation and it is expounded by Peter here:

1Peter1:18 Forasmuch as ye know that ye were not redeemed with corruptible things, as silver and gold, from your vain way of life received by tradition from your fathers;
19 But with the precious blood of Christ, as of a lamb without blemish and without spot:
20 Who verily was foreordained before the foundation of the world, but was revealed in these last times for you,
21 Who by him do believe in God, that raised him from the dead, and gave him glory; that your faith and hope might be in God.
22 Seeing ye have purified your souls in obeying the truth through the Spirit to a sincere love of the brethren, see that ye love one another with a pure heart fervently:
23 Being born again, not of corruptible seed, but of incorruptible, by the word of God, which liveth and abideth for ever.
24 For all flesh is as grass, and all the glory of man as the flower of grass. The grass withereth, and its flower falleth away:

Here born again is the word:

annagenao and specifically means : to produce again, be born again, born anew

The word for iniatiate is mueo from the word mysterion. The word egkainizo referrs to a ritual of renewal or consecration or initiation. Neither of these is used in any of Jesus teachings.

It is very helpful to read things in their original languages because then we don't lose the sense in the confusion of transliteration.

>>Matthew 3:11 (John the Baptist speaking)
I indeed baptize you with water unto repentance. but he that cometh after me is mightier than I, whose shoes I am not worthy to bear: he shall baptize you with the Holy Ghost, and with fire.

This is a baptism unto repentance!! Not an initiation rite. John uses the word baptizo which means to dip into water to clean or wash. Jesus uses the word baptisma which has connotations of of calamities and afflictions with which one is quite overwhelmed ..... :

Romans 6:3 Know ye not, that so many of us as were baptized into Jesus Christ were baptized into his death? {were: or, are}
4 Therefore we are buried with him by baptism into death: that as Christ was raised from the dead by the glory of the Father, even so we also should walk in newness of life.
5 For if we have been planted together in the likeness of his death, we shall be also in the likeness of his resurrection:
6 Knowing this, that our old man is crucified with him, that the body of sin may be destroyed, that henceforth we should not serve sin.
7 For he that is dead is freed from sin. {freed: Gr. justified}
8 Now if we are dead with Christ, we believe that we shall also live with him:
9 Knowing that Christ being raised from the dead dieth no more; death hath no more dominion over him.
10 For in that he died, he died to sin once: but in that he liveth, he liveth to God.

(baptizo used above)

Here again, Jesus is referring to His crucifixion:
Luke 12:50 But I have a baptism to be baptized with; and how am I constrained till it shall be accomplished!

and here:

Mark 10:38 But Jesus said to them, Ye know not what ye ask: can ye drink of the cup that I drink of? and be baptized with the baptism that I am baptized with?

This baptism is the suffering. The filling with the HolySpirit is the annointing with God's Spirit. They are two different things. The second only took place after Jesus ascended to heaven:

John 14:26 But the Comforter, who is the Holy Spirit, whom the Father will send in my name, he shall teach you all things, and bring all things to your remembrance, whatever I have said to you.

The birth of the Spirit only began after Jesus ascended to heaven .... not while He was on earth.

There are many other verses where Jesus speaks of this, but I used the one in John because you seem to use John as an authority. If you read the whole of John (to get the context right) you will find that it has nothing to do with initiation.

Blessings
Banah