"So help me God" –– What does it mean?

I recently came across this snippet from the William & Mary Law Review (1992) vol34/iss1/13: Religion-Plus-Speech: The Constitutionality of Juror Oaths and Affirmations Under the First Amendment by Jonathan Belcher:

The phrase "so help me God," a popular component of grand and petit juror oath statutes, is actually an abbreviated form of the oath, "So may God help me at the judgment day if I speak true, but if I speak false, then may He withdraw His help from me." White, supra note 30, at 379-80 n.10.

When I tracked down Thomas R. White, Oaths in Judicial Proceedings and Their Effect Upon the Competency of Witnesses, 51 U. PA. L. REV. 380, 380 (1903), this is what White says in a footnote:

In some instances the words of imprecation were added to oaths given in the Old Testament. Ruth to Naomi says, "The Lord do so to me and more also, if aught but death part thee and me." Ruth i. 17. The same words were used by Eli to Samuel when he was adjuring him to speak the truth. I Sam. iii. 17.

Many of the later Christian oaths contained express imprecations of God's vengeance for false swearing. Selden says that among the Spaniards the oath used in making a covenant concluded thus, "If I first designedly fail of this oath on that day, ye powers above torment my body in this life and my soul in the next with horrid tortures. Make my strength and my words fail. In battle let my horse and arms and spurs and subjects fail me when need is the sorest." II, II.

The words now used, "So help me God," at present little understood, are but a shorter way of expressing the same imprecation. When this precise form was first adopted is uncertain. Tyler says that he has been unable to find any oath in England which does not contain it or a similar phrase. The meaning of the phrase is, "So may God help me at the judgment day if I speak true, but if I speak false, then may He withdraw His help from me." Tyler, C. III.

So far, I haven't been able to identify this "Tyler, C. III."

I did find: James Endell Tyler - Oaths: their origin, nature, and history (1834). From what can be seen, J.E. Tyler avoids any reference to "So help me God" within what he provides as an enlightened definition of an oath:

"An oath is an outward pledge given by the juror that his attestation [or promise] is made under an immediate sense of his responsibility to God."

In a following footnote he adds:

* I have never found either the definition of an oath, or the form of an oath implying the imprecatory clause, acquiesced in by the early Christians. The ages when the most dreadful imprecations were used, and a multiplication of them was relied upon as a greater security for the truth, were the ages of religious darkness and corruption.

On page 12 he explains:

Phillips, indeed, in his beautifully luminous and able Treatise on the Law of Evidence, seems to take it for granted, that "our law (like that of most civilized nations) requires a witness to believe, not only that there is a God and a future state of rewards and punishments, but also that, by taking the oath, he imprecates upon himself, if his evidence is false*," But as we shall often be reminded, this view is neither universally taken, not is it insisted upon by the jurisprudence of all European countries. M. Merlin having stated that, by ancient law, the members of two different religions in France had the privilege of taking their own peculiar oath, Jews and Anabaptists, observe the latter, "their religion only allows them to say 'Yes," to the form of oath administered by the judge, and forbids them to lift their hand, because they believe that would be to provoke the Lord from the highest heaven; an act which, according to them, would be an impiety more fitted to destroy the credit of him who should be guilty of it, than to merit confidence."

* Phillips, c.3

On page 57 he explains:

I would again observe, that whilst the mode now universally adopted among us, is imprecatory; –– the invoking of God's vengeance in case we do not fulfill our engagement to speak the truth, or perform the specific duty, "So help me God," –– I am not aware of the form being sanctioned by the words or the example of Christ himself, or any of his Apostles, or of those whom we regard as the most approved models of primitive Christianity.

In contrast to the deep-rooted understanding of an impracatory oath as just shown, here's what American Creation blogger Brian Tubbs wrote back on 12/29/2011:

When a President-elect says "so help me God" after the oath, he is simply adding his personal sentiment. He's not writing law. He is asking God to help him fulfill his legal obligations.

So says Brian Tubbs. But look, even if a president would of his own accord choose to omit those non-biblical and extra-constitutional four words, it appears that he would still be doing so at his everlasting peril. See The Everything U.S. Constitution Book: An easy-to-understand explanation of the foundation of American government (2011), page 53-sidebar, by Ellen M. Kozak:

Legend has it that George Washington added the words "so help me God" to the end of the oath, but there is no more proof of that than the legend of the cherry tree. As to whether all other presidents have added it since, it is likely that Thomas Jefferson* did not. However, now that inaugurations are broadcast, presidents omit it at their peril, since the public expects it.

* See my American Creation blog of 6/21/2008 where Barbara Oberg, General Editor of The Papers of Thomas Jefferson offers her comment about "TJ".