As to the divine names, Elohim, אלהים is the common name for God, Him with whom we have to do; hence for all who are viewed in this place by man, or represent Him who is rightly so viewed, as judges in Israel (Ex. 21), or angels. (Psa. 8) Of course there is but one true God, but gods many amongst men: but hence, in Elijah's history, "Jehovah is the Elohim, Jehovah, he is the Elohim." It is the word in universal use for God as such. But constantly, when Elohim is distinctly used for the one true God, the article is added Ha Elohim. Eloah is the singular of Elohim.
El (אל) is the strong or mighty one, who stands, so to speak, by His own power. Hence we have El-Elohe-Israel, El (God, the mighty one) אל, the God of Israel. El and Eloah are constantly used in Job. (אל chap. 5:8; 8:13, 20; 9:2; 12:6; 13:3, 7, 8; 15:4, 13, 25; 16:11, etc. אלוה 3:4, 23; 4:9; 5:17; 6:4, 8, 9; 9:13; 10:2; 11:5, 6, 7; 12:6; 15:8, etc.) It is said to be Aramean. So we have in Daniel El Elion (most high God).
I can say nothing satisfactory to my own mind as to Jah. That it is used as an ancient poetic name for Jehovah is clear, as in Hallelujah. So in Ex. 15 "my strength and my song is Jah." If you look into most dictionaries, you will find it stated to be a shortening of Jehovah. But then I find them used together, as Isa. 26:4,4Trust ye in the Lord for ever: for in the Lord JEHOVAH is everlasting strength: (Isaiah 26:4) "In Jah Jehovah is everlasting strength." So Isa. 12:2,2Behold, God is my salvation; I will trust, and not be afraid: for the Lord JEHOVAH is my strength and my song; he also is become my salvation. (Isaiah 12:2) "My strength and my song is Jah Jehovah." It is found in Psa. 68 where a ב precedes, translated "by His name Jah" in English, but which may be doubted, though a name of holy song and praise at any rate.
Jehovah is God's name of dealing and relationship with men, specially with Israel, derived (I suppose) from חיח, to exist; and practically translated "who is and was and is to come," not, "who was, and is," which is true, but "is" (exists, that is, in Himself eternally) "and was and will be" in past and future true. Hence He is one who having spoken makes good. "Thou art the same and thy years shall not fail." Hence we have in Gen. 1, "Elohim," the Creator; in Gen. 2, "Jehovah Elohim," because the relationship of God with men is spoken of; for there it is not His place simply over creatures as such connected with God, but all His various relationships: how Adam was placed, warned, and dealt with, his wife's place with him, creatures' subjection, etc. These words, Elohim and Jehovah, are never confounded in scripture. The senseless scissors' distinction of rationalists shows only their want of looking intelligently into the written word of God. God is God as such; Jehovah, One who enters into relationship with His people and with men.
There is another name אלשךי by which God revealed Himself, that is to Abraham and the patriarchs, El Shaddai. See Ex. 6:3,3And I appeared unto Abraham, unto Isaac, and unto Jacob, by the name of God Almighty, but by my name JEHOVAH was I not known to them. (Exodus 6:3) where Elohim takes specially the name of Jehovah as the God of Israel. These two names are beautifully brought out in 2 Cor. 6:1818And will be a Father unto you, and ye shall be my sons and daughters, saith the Lord Almighty. (2 Corinthians 6:18) to take the name of Father with us. "I will be a Father and ye shall be my sons and daughters," says Jehovah Shaddai, the God who was the one to Israel, the other to Abraham. In Gen. 2:33And God blessed the seventh day, and sanctified it: because that in it he had rested from all his work which God created and made. (Genesis 2:3) it was of all importance to connect Jehovah, Israel's national God, with the only one creator God. So in Ex. 9:3030But as for thee and thy servants, I know that ye will not yet fear the Lord God. (Exodus 9:30) the God of the Hebrews, whose name was Jehovah, is declared to be Elohim; Pharaoh would not yet fear Him. Otherwise Jehovah is a name, Elohim a being: only Jehovah is Elohim, but the former a personal name. אלעולם El Olam is the everlasting God. See Gen. 21:3333And Abraham planted a grove in Beer-sheba, and called there on the name of the Lord, the everlasting God. (Genesis 21:33).
Elion (אלעליון) is the Most High God. This is a fourth name God takes in connection with men; His millennial name above all idolatrous gods and demons and all power, and then said to be "possessor of heaven and earth." Hence, when Nebuchadnezzar is humbled, after being a beast till seven times had passed over him, he owns the God of the Jews to be the Most High God. So in Dan. 7; but not when connected with saints: there it is plural (Elionin) and refers, I believe, to the high or heavenly places. (Vers. 18, 22, 25, 27.)
אהיה Ehejeh in Ex. 3 is merely the abstract tense in Hebrew, and "I am that I am" I believe to be right enough. [Some take it as "I will be that I will be."]
Adonai (אדני) is simply "Lord" (in the plural of majesty as is said), but hence, I believe, is used for Christ, exalted as man, but Jehovah withal, as Psa. 110:55The Lord at thy right hand shall strike through kings in the day of his wrath. (Psalm 110:5). It is also Adonai in Psa. 2:44He that sitteth in the heavens shall laugh: the Lord shall have them in derision. (Psalm 2:4); Isa. 6:1, 81In the year that king Uzziah died I saw also the Lord sitting upon a throne, high and lifted up, and his train filled the temple. (Isaiah 6:1)
8Also I heard the voice of the Lord, saying, Whom shall I send, and who will go for us? Then said I, Here am I; send me. (Isaiah 6:8); Dan. 9:1717Now therefore, O our God, hear the prayer of thy servant, and his supplications, and cause thy face to shine upon thy sanctuary that is desolate, for the Lord's sake. (Daniel 9:17).
There is another word which, though it may be used as an attributive, can hardly be excluded from being a name of God. חוא Hu, Atta Hu, "thou art the same," the unchangeable One (see Deut. 32:3939See now that I, even I, am he, and there is no god with me: I kill, and I make alive; I wound, and I heal: neither is there any that can deliver out of my hand. (Deuteronomy 32:39)); "I am He, the same and besides me no god." I am He, חוא: Psa. 44:44Thou art my King, O God: command deliverances for Jacob. (Psalm 44:4); Isa. 41:4; 43:10, 134Who hath wrought and done it, calling the generations from the beginning? I the Lord, the first, and with the last; I am he. (Isaiah 41:4)
10Ye are my witnesses, saith the Lord, and my servant whom I have chosen: that ye may know and believe me, and understand that I am he: before me there was no God formed, neither shall there be after me. (Isaiah 43:10)
13Yea, before the day was I am he; and there is none that can deliver out of my hand: I will work, and who shall let it? (Isaiah 43:13); Jer. 5:1212They have belied the Lord, and said, It is not he; neither shall evil come upon us; neither shall we see sword nor famine: (Jeremiah 5:12). It is in the sense of the immutable existing One, which is true of God only, as Psa. 102:2727But thou art the same, and thy years shall have no end. (Psalm 102:27) cited in Heb. 1, ὁ αὐτός.
Though the Psalms afford in the most interesting way the difference of the use of God and LORD, I just refer to Gen. 7:1616And they that went in, went in male and female of all flesh, as God had commanded him: and the Lord shut him in. (Genesis 7:16). God commanded him, it was Elohim's order; and Jehovah, the personal God that cared for him, not merely the divine being, shut him in. The scissors must be very small and fine that cut this into two documents, while the Lords mind shines out with the deepest beauty and interest to those that have eyes to see. So in chapter 8:21 we have "Jehovah smelled a sweet savor," because it was a personal relationship and dealing with men. All the rest of this part is Elohim, God as God dealing with a subject world in the flood, and sparing, as such, Noah and his family. In chapter 9:26 Jehovah comes in again in evident relationship. As to all these, readers have only to take an "Englishman's Hebrew Concordance" and seek the passages where these various names occur, generic, personal, official, or compound. It will not be lost labor, nor anything which is an inlet to the divine mind—God's revelation of Himself.