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, "In Jah Jehovah is everlasting strength." So Isa. 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, where Elohim takes specially the name of Jehovah as the God of Israel. These two names are beautifully brought out in 2 Cor. 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. 2:3 it was of all importance to connect Jehovah, Israel's national God, with the only one creator God. So in Ex. 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. 21:33.
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. 110:5. It is also Adonai in Psa. 2:4; Isa. 6:1, 8; Dan. 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. 32:39); "I am He, the same and besides me no god." I am He, חוא: Psa. 44:4; Isa. 41:4; 43:10, 13; Jer. 5:12. It is in the sense of the immutable existing One, which is true of God only, as Psa. 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. 7:16. 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.