You asked a series of questions about the Holy Spirit arising from a passage in Acts 2.
"Is the gift The Holy Spirit?
Is the gift the Holy Ghost?
Is the gift eternal life based on revelation of who Jesus is?
Why do I asked?
KJV: For the promise is unto you, and to your children, and to all that are afar off, even as many as the Lord our God shall call.
Well what promise? Where is this particular promise mentioned in the old testament?"
The parts of the passage I underlined answer your questions pretty clearly, I think. The "promise" is the Holy Spirit, who was sent after the ascension of Christ as his "replacement" (so to speak).
John 16:7
7 "But I tell you the truth, it is to your advantage that I go away; for if I do not go away, the Helper will not come to you; but if I go, I will send Him to you.
In Acts 2, Peter himself cited where in the OT the promise of the Holy Spirit was issued:
Acts 2:16-17
16 But this is what was uttered through the prophet Joel:
17 “‘And in the last days it shall be, God declares, that I will pour out my Spirit on all flesh...
(See: Joel 2:28-32)
Who has received the promise of the Holy Spirit?
All those who by faith in Christ as Savior and submission to him as Lord are made "temples" of the Holy Spirit in whom the life of Christ is given to them. (Titus 3:5; Romans 8:9-16; John 16:7-14; 1 John 4:13, etc.)
What is the promise of the Holy Spirit? See above.
What is the gift of the Holy Spirit? The Holy Spirit himself; he is the gift.
??? It is a fact that the Holy Spirit did come and go from people prior to the Atonement. Samson is a good example. Or Jeremiah. Or Daniel. Or Elijah. None of these men had a constant indwelling of the Holy Spirit.
In Acts 2, which is what I assume you mean by "here," we know the Holy Spirit did remain on the disciples of Jesus permanently because they say as much in their various letters, making his evident presence within the believer the chief means of discerning whether or not a person was genuinely saved.
Romans 8:9
9 You, however, are not in the flesh but in the Spirit, if in fact the Spirit of God dwells in you. Anyone who does not have the Spirit of Christ does not belong to him.
1 John 4:13
13 By this we know that we abide in him and he in us, because he has given us of his Spirit.
2 Corinthians 1:21-22
21 Now He who establishes us with you in Christ and anointed us is God,
22 who also sealed us and gave us the Spirit in our hearts as a pledge.
Hebrews 13:5
5 Make sure that your character is free from the love of money, being content with what you have; for He Himself has said, "I will never desert you, nor will I ever forsake you,"
It is the difference between intellectually agreeing that a chair can hold you and actually sitting in it.
Yes. But he is the refreshing himself. His presence within, the life and light that he is inside of you, refreshes, or, in the parlance of Paul, "regenerates" and "renews." (Titus 3:5)
There is no difference. Certainly, Paul offers nothing in this remark to Timothy to suggest that he thought there was.
I see nothing in God's word to cause me to "suspect" as you do.
Not sure what your point is here... The Holy Spirit "pours out" of himself what he is: life, power, love, joy, peace, patience, gentleness, goodness, etc. (Acts 1:8; Romans 8:10; Galatians 5:22-23)
The Holy Spirit is the Third Person of the Triune Godhead.
Just saying, "Not so" doesn't make it so. Show how my understanding is in error; don't simply assert that it is and then offer an alternative view. Such a response is merely dismissive which is hardly a refutation of what I pointed out.
Description does not equal prescription. It is a deeply fallacious way of understanding Scripture to take a description of an event in it and make that event prescriptive for believers when no such prescription is actually issued from the event. Nowhere in the Bible are we told that the event of Christ's baptism by John, and the supernatural response from God that attended it, we ought to expect will be God's response to us, too. We have only the description of the event, nothing more, no added "And so it will be for all who follow Christ," or some such statement. It is a sure route to false notions about spiritual living to force into the record of the event an expectation for all believers.
There is no separate "anointing" of the Holy Spirit following his indwelling the born-again person. To have him come to dwell within is to receive all that God can impart spiritually of Himself in the fullest degree. What remains is for the believer to live in daily, persistent submission to God (Romans 6:13-21; Romans 8:14; Romans 12:1; James 4:6-10; 1 Peter 5:6) that the Spirit may freely enter and transform every "room" in the "house" of the believer's life, conforming him/her to the character of Christ (Romans 8:29) and making of them a "vessel sanctified and prepared for the Master's use" (2 Timothy 2:21).
Because of the hardness of their hearts, many will not believe. In the record of Acts when some were saved and others not, those who were not had been "suppressing the truth in unrighteousness" (Romans 1:18-22) and so had grown deaf, blind and hard to the saving truth of the Gospel (Hebrews 3:13). They also did not want to "come to the light lest their evil deeds should be exposed" (John 3:20) because they "loved the darkness rather than the light" (John 3:19). This is always, at bottom, why sinners refuse to be saved.
Pour out AND in. You're forming a false dichotomy here, I think, artificially making "pouring out" and "pouring in" mutually-exclusive of one another. They're not. The Holy Spirit was poured out on the disciples in Acts 2, taking up permanent residence within them, making of them his "temples" (1 Corinthians 6:19-20; Romans 8:9-16).
If it was important to know, God would have seen to it that their words were recorded.
There was no overt, external event accompanying the Ethiopian eunuch's salvation. He did not speak in tongues, or roll about on the ground in a fit of convulsive hysteria (aka "slain in the Spirit"). Lydia, a seller of purple, was saved but she did not speak in tongues, or go out and perform a miraculous healing, or make a prophetic declaration. The Philippian jailer was saved but there was no overt manifestation of the Spirit when he was. And so on.