Is Heaven the same place as the New Earth?

In the Bible the word ‘heaven’ or ‘heavens’ have different meanings, as in the example of Noah and the flood where “the windows of heaven were opened” and it rained:

Genesis 7:11-12 King James Version (KJV)
11 In the six hundredth year of Noah’s life, in the second month, the seventeenth day of the month, the same day were all the fountains of the great deep broken up, and the windows of heaven were opened.
12 And the rain was upon the earth forty days and forty nights.

The Bible says “all the high hills that were under the whole heaven, were covered”:

Genesis 7:19 King James Version (KJV)
19 And the waters prevailed exceedingly upon the earth; and all the high hills, that were under the whole heaven, were covered.

Another heaven or second heaven in the Bible is used in the sentence “as the stars of heaven”:

Exodus 32:13 King James Version (KJV)
13 Remember Abraham, Isaac, and Israel, thy servants, to whom thou swarest by thine own self, and saidst unto them, I will multiply your seed as the stars of heaven, and all this land that I have spoken of will I give unto your seed, and they shall inherit it for ever.

This heaven or second heaven is also called “the heaven of heavens” referring to space, and “with all their host” refers to the stars and planets:

Nehemiah 9:6 King James Version (KJV)
6 Thou, even thou, art Lord alone; thou hast made heaven, the heaven of heavens, with all their host, the earth, and all things that are therein, the seas, and all that is therein, and thou preservest them all; and the host of heaven worshippeth thee.

Lastly there is a “third heaven”, where Paul “knew a man in Christ above fourteen years ago” that was “caught up to the third heaven”, and “was caught up into paradise”:

2 Corinthians 12:1-4 King James Version (KJV)
1 It is not expedient for me doubtless to glory. I will come to visions and revelations of the Lord.
2 I knew a man in Christ above fourteen years ago, (whether in the body, I cannot tell; or whether out of the body, I cannot tell: God knoweth;) such an one caught up to the third heaven.
3 And I knew such a man, (whether in the body, or out of the body, I cannot tell: God knoweth;)
4 How that he was caught up into paradise, and heard unspeakable words, which it is not lawful for a man to utter.

Before the New Earth is made, the righteous have already “lived and reigned with Christ a thousand years” in heaven:

Revelation 20:4 King James Version (KJV)
4 And I saw thrones, and they sat upon them, and judgment was given unto them: and I saw the souls of them that were beheaded for the witness of Jesus, and for the word of God, and which had not worshipped the beast, neither his image, neither had received his mark upon their foreheads, or in their hands; and they lived and reigned with Christ a thousand years.

After the “thousand years” in heaven, the new heaven and new earth is made which becomes heaven where “God himself shall be with them, and be their God”:

Revelation 21:1-7 King James Version (KJV)
1 And I saw a new heaven and a new earth: for the first heaven and the first earth were passed away; and there was no more sea.
2 And I John saw the holy city, new Jerusalem, coming down from God out of heaven, prepared as a bride adorned for her husband.
3 And I heard a great voice out of heaven saying, Behold, the tabernacle of God is with men, and he will dwell with them, and they shall be his people, and God himself shall be with them, and be their God.
4 And God shall wipe away all tears from their eyes; and there shall be no more death, neither sorrow, nor crying, neither shall there be any more pain: for the former things are passed away.
5 And he that sat upon the throne said, Behold, I make all things new. And he said unto me, Write: for these words are true and faithful.
6 And he said unto me, It is done. I am Alpha and Omega, the beginning and the end. I will give unto him that is athirst of the fountain of the water of life freely.
7 He that overcometh shall inherit all things; and I will be his God, and he shall be my son.