Having the eternal life of God in us is not the same thing as the Holy Spirit indwelling in us. The Holy Spirit is the 3rd Person of the Godhead, whereas eternal life is life eternal which God has. They are separate things.

God's life is eternal. Yes, the Holy Spirit is the life of the Father and the Son and is eternal. Technically speaking, eternal life is not the Holy Spirit. The 3rd Person is not some impersonal concept such as eternal life, but He is one who holds council in the Trinity of the Godhead, just as the Father is one member and the Son is one member of the Triune God.

When we are born again, yes, we receive the Holy Spirit because we have been given a new life, eternal uncreated life, which the Holy Spirit can indwell in our spirit. Therefore it is indeed wrong to say "When we are born again from above we receive the Holy Spirit (eternal life or the uncreated life) into our human spirit" since the Spirit has eternal life, but He is not eternal life.

God's life is eternal life because the Holy Spirit, which is God's life, has eternal life! We can not say "God's life is eternal life because the Holy Spirit is eternal life." God is not a thing. And those who have eternal life, God's [Own] life, we are still not God the Holy Spirit, God's Own life. Do you see there are precisely two different usages of the term, God's life?

People say God is love, eternally existing, spirit, omnipotent, omniscient, omnipresent, truth, in Christ, and many other things. All these things describe God. Each make up a component of God's characteristics. But to say God the Spirit is eternal life is wrong. Can you sense that in your spirit? We can say God is eternal and we can say God is life or the source of life to describe God. Yet we can not say God is eternal life. We can only say God has eternal life. I don't know how to explain it any better.

The reason why these terms "Holy Spirit" and "eternal life" and "Christ in you" and "Christ lives in me" are different are not because they are a "different aspect" being shown each time of the same thing, but of different things. Christ is in me by the Holy Spirit and lives in me and changing me in the abode of eternal life of my new spirit where the Holy Spirit indwells.