I just came across this rather helpful article by Ryan Conrad Davis and Paul Y. Hoskisson:
"Usage of the Title Elohim," Religious Educator 14, no. 1 (2013): 109–127
Here is the conclusion:
As detailed above, Church members prior to the authoritative clarifications of the early twentieth century often used elohim and Jehovah interchangeably and inconsistently, much the same way they are used in the Hebrew Bible. Like much of the Christian world of the nineteenth century, Latter-day Saints did not always distinguish between Jehovah, God the Father, the God of Israel, elohim, or simply God. However, the flexibility of use and at times the ambiguous phrasing of the nineteenth century that reflected general American usage and served the general Christian world well, fell short of the precision that the restoration of the gospel brought to Latter-day Saint understanding of the Godhead.
It is remarkable that early Latter-day Saints used the name Jehovah in reference to both God the Father and to his Son. Equally interesting is that elohim seems to have been used by Latter-day Saints for both God and gods, exactly as it is used in the Hebrew Bible, that is, as both a singular and a plural noun, a proper name and a common noun. Officially, this practice ended in 1916.
And finally, a word of caution here is appropriate. Since the modern Latter-day Saint usage of Jehovah and Elohim was not taken from the Hebrew Bible, it can create misunderstandings if imposed upon the Hebrew scriptural account. Thus if we try to exclusively assign actions to different members of the Godhead based on which divine name is used in the Hebrew Bible, the result, in many instances, will be chaos. Additionally, D&C 20:28 states that “Father, Son, and Holy Ghost are one God” (see also 2 Nephi 31:21; Alma 11:44; 3 Nephi 11:27, 36; Mormon 7:7). In this same vein, Elder Bruce R. McConkie once said that “most scriptures that speak of God or of the Lord do not even bother to distinguish the Father from the Son, simply because it doesn’t make any difference which God is involved. They are one. The words or deeds of either of them would be the same words and deeds of the other in the same circumstance.” Therefore, the issue of which name or title is assigned to which member of the Godhead is not one that Latter-day Saints should be overly concerned with. But it is helpful to know that the meaning of a word such as elohim is not always the same in all times and in all places.
As an aside, Paul Hoskisson wrote a very important volume a few years ago now which is available for free online: