The words אֱלֹהֵ֨י, וֵֽאלֹהֵ֤י and אֱלֹהֵ֖י in Genesis 31:53 are the same noun, elohim in three slightly different forms:
- אֱלֹהֵ֨י, elohei is the noun elohim in associative form, "the deity of"
with a readers mark indicating a rising tone at the beginning of a clause
- וֵֽאלֹהֵ֤י, weylohei, is the same as above with the conjunction "and" prefixed, and a readers mark indicating a high penultimate tone before the end of a clause pause tone
- אֱלֹהֵ֖י, elohei, is the same as (1) above, but with a readers mark indicating falling tone before a major clause break, an atnachta.
The meaning is "the deity of" or "the gods of", in this context. This is not a name of God, the LORD, as the words are spoken by Lavan, who does not worship God and who has come to recover his idols from Jacob.
In this verse Lavan expresses the sincerity and severity of the oath between himself and Jacob by referring to the gods of Abraham, and the gods of Nahor. The and before "the gods of Nahor" shows that Lavan sees them as being different, though he refers to them in the same breath, making no distinction between them, even though we, the readers, already know by this point in the narrative that there is world of difference between the God of Abraham and the gods of Nahor. It is this lack of distinction that indicates that Lavan is not referring to "The God of Abraham" but rather to "the gods of Abraham". At this point the verse is ironic, causing us to raise an eyebrow. But then Lavan heightens the seriousness by using a common Semitic form of oath, invoking the father - not only the gods of Abraham and Nahor, but their daddy's gods as well! The bathetic humor at this point in the verse is not lost on native Hebrew or Arabic speakers. The text is laughing at Lavan.
The Names of God translation seems to have misunderstood this throughout the verse, used the capitalized transliteration "Elohim", a moniker for God, where the Hebrew says elohei, and also eliding the elohei of Nahor into the elohei of Abraham leaving only two eloheis where the MT has three.
The Holman translation is better but interprets the first אֱלֹהֵ֨י that is associated with Abraham as אֱלֹהֵ֨י in the sense of a name of God, possibly following the classical Jewish interpretation brought by Rashi which, IMHO is a mistake. Indeed, "the God of Abraham" is a common moniker for God, but not in this context where the speaker is Lavan the idol worshiper who does not distinguish between God and gods.
The word elohim, is a plural form that can have either singular or plural meaning in Hebrew, whose meaning in English is best understood as the singular "deity". It can refer either to either
- the deity or deities in a general sense that are the object of someones worship
- God, the LORD, when the identity of the speaker or the context indicates that this refers to God
When referring to God, Elohim, "the Deity", is a moniker, like "the God of Abraham" or "the Fear of Issac" (see the end of Genesis 31:53), and not a proper name like the tetragrammaton, which is often rendered in English as "the LORD" or YHVH, Jehovah and similar.
In most instances in the OT the context adequately distinguishes between elohim, the deity, and Elohim, the Deity. In some verses it is not clear. My own translation of Genesis 31:53 would be:
"The gods of Abraham and the gods of Nahor, [even] the gods of their fathers will judge between us!", but Jacob swore in the name of the Fear of his father, Isaac.
Note that I change the MT word order slightly, as I believe that the subject of "will judge between us" is the subsequent "the god of their fathers" combined with the the previous "the gods of Abraham and the gods of Nahor". This parsing is consistent with both of the translations quoted in the OP.