Why do some seek 'two beginnings' in scripture?

Since you mentioned that you are interested in all perspectives - Old Earth Creationism (OEC) as well as Young Earth Creationism, I believe that I can explain at least one popular variant of Old Earth Creationism concerning your question. This would be the OEC perspective of Hugh Ross and the scholars at reasons.org. They take the same perspective that both @rhetorician and @Mike Borden explained, namely that in John 1:1 we are really talking about the eternal existence of Jesus the Creator while in Genesis 1 we are talking about the physical creation of the cosmos, including earth and life on earth. Ross would go even further to assert that since Einstein's General Theory of Relativity points to the beginning of space and time itself, that this is all the more reason to take John 1:1 as from the point of eternity, beyond the space-time continuum that makes up our universe. Also, concerning the earth being "without form and void", this would correspond to what astrophysicists have found about the beginning of the earth being a chaotic mass of rocks until it was formed in stages to the beautiful planet that we have today. All of the "coincidences" that made this transformation take place were highly improbable even according to scientist non-believers. And Ross makes many calculations and shows specific instances concerning these improbable events (hence the difficulty of finding a similar planet to ours that could sustain carbon based life even among the numerous planets that have been found so far). So, to me this OEC perspective of the scholars at reasons.org are consistent with your original intuitions of your youth and yet provide a very coherent harmony with the majority of findings of science as well.

Gap theory (Ruin Reconstruction Theory) is an attempt to reconcile those who wish to take the Bible literally with a scientific view of a very old Earth.

The universe—heaven and Earth—was originally ("in the beginning") created aeons ago; life flourished for millions or billions of years. But this world (perhaps just Earth and not the entire universe) grew to be evil, and God destroyed it in a gigantic cataclysm. Earth became "without form and void" as a result of this destruction. (Gap theorists hold that the verb in the second verse is more accurately translated as became or had become rather than as was. The familiar six-day creation—a re-creation really—then followed, mere thousands of years ago, upon the ruin and chaos of this ancient former world.

Gap theory advocates, by this maneuver, are able to reconcile the scientific evidence for an old Earth and universe and for life itself. They, just as much as the young-Earth creationists, reject evolution; to them, the re-creation six thousand or so years ago was not entirely ex nihilo (although humans may have been created out of nothing) but was certainly by divine fiat. Therefore, although they differ markedly from "strict" creationists regarding the age of Earth, their antievolution attitudes and arguments are virtually identical. - [NCSE Creation/Evolution Journal, vol.8, no.3, Fall 1988 1

This insertion of a vast expanse of time between the first two verses of Genesis is undertaken without any reference at all to the first verse of John's gospel.

A similar attempt to reconcile Scripture with modern science posits a day/age scenario in which each of the six days of creation are not literally 24 hour periods but are, instead, periods of hundreds of millions or even billions of years. Again there is no mention of John 1:1 in the formulation of this theory.

Difficulties in or refusal to accept the "In the beginning" of Genesis and John as referring to the same instance are usually driven by a theological need to strip Jesus, the Son of God, of His pre-incarnate divinity. A common theme includes the notion that "the Word" was God's first, and only, direct creation after which everything else was created by God through the Word.

In this paradigm if Genesis 1:1 is the beginning of the creation of all that there is then John 1:1 has already occurred and if Genesis 1:1 can be widened to implicitly include the creation of the Word then "in the beginning" of John 1:1 is not equivalent to Genesis 1:1 but merely one aspect which is less than the whole.

Apples and oranges, my friend. Apples and oranges.

John 1:1 and following concern the eternality of the Son of God, the Logos, who existed in eternity, but who became flesh in obedience to the Father and out of his great love for us, his image-bearers.

Moreover, while John 1:1 and following paint a co-mixture of both spiritual (viz., the Word) and material (viz. "all things" and "became flesh"), Genesis 1:1 and following concern primarily the material.

Put differently, there are not two beginnings but two different uses of the word beginning. The apostle John's use of the word is his attempt, aided by the Holy Spirit, to describe two realities. The first is an eternal reality to which no measurement of time can be attached. It is the eternality of God and the Word of God, the Logos.

The question arises as a paradox, however, concerning time versus no time. Eternity (no time) cannot be measured, but time can. Without the eternal, there would be no material, no temporal, no time. The spiritual and the eternal precede the material and temporal. That's John 1, writ large.

Genesis 1:1 and following concern primarily the physical, material, and temporal. Moses wrote for the benefit of the people he led out of captivity and servitude into freedom and theocratic nationhood. The Hebrews needed to be reminded of the creatorial power of YHWH. They could take comfort in his watchful care over them because he had proved himself to be both the Almighty Creator and a miracle worker (e.g., he parted the Red Sea). Moreover, YHWH had covenanted with their forebears, beginning with Abram (later, Abraham) to make them a great nation under God, as well as a witness and a blessing to the entire world (see Genesis 12:3).

As for your questions about two creations, or one creation and one re-creation, frankly I have no answers for you. I do, however, have a question for you. Why did Moses describe the earth the way he did in Genesis 1:2?

Now the earth was formless and empty, darkness was over the surface of the deep, and the Spirit of God was hovering over the waters (NIV).

To me, that verse sounds as though there needed to be a re-creation of something that was already in existence. However, what seems to have been already in existence was an earth that was formless, empty, and dark. The God I worship is a God of order, not chaos (see 1 Corinthians 14:33). The earth as Moses describes it in Genesis 2:1 is in complete chaos. Why? How did it get that way?

Again, I have no answer for you. Perhaps some brave soul will be able to provide a biblical answer that satisfies you.