I/we shall, and you/he/she/it/they will, expresses simple futurity – something that is clearly going to happen, all other things being equal. “The sun will rise tomorrow morning, and I shall awaken.”
I/we will, and you/he/she/it/they shall, expresses determination – “I will prevail, no matter what the odds” or “Despite our poverty, you shall have the opportunity to finish college.”
Amusingly, what screwed this up was somebody using it correctly, but to make a special point.
When Douglas MacArthur was ordered to leave the Philippines, for which he had a special place in his heart, and which were rapidly being overrun by the Japanese, he thought that the wise way to promise their eventual liberation was to avoid using the determination form and state as his conviction in the simple future: “I shall return.” Unfortunately, that got taken as a statement of determination, and about defeated the precisionist usage of shall/will, so that they now are somewhat interchangeable, with “shall” in all persons expressing something more of a determination.
As for “should,” it’s the conditional of “shall” — “If I should die before I wake, I pray the Lord my soul to take.” It’s used exactly parallelly with “would.” It’s distinct from a quite different use of “should” as a synonym of “ought” – “I should get my homework done, but I think I’ll surf the Internet instead.”
What Paul in Saudi said is technically wrong, but makes the point that in precise language and expressing simple futurity, you use shall in the first person and will in the second and third. What he missed is that when expressing intentionality, you reverse the usage.