We are dealing with a different way of counting time in the Bible than we do today. I mean, the fact that the 24 hour "day" started as a sunset is totally foreign to us outside of Jewish and other religious reckoning handed down. And then you have the way that "part of a day" was counted as a full day (ie a day and a night) according to the Talmud.
I indicated 13 examples in http://historical-jesus.info/77.html
showing the way ancients were counting days was the same as today:
1) Evidence about "after three days" is the fourth day and "the third day" is "after two days":
A) The fourth day (and NOT the third) immediately follows a period of three days, according to ancient customs:
a) Jdg 19:4-5a "Now his father-in-law, the young woman's father, detained him; and he stayed with him three days. So they ate and drank and lodged there.
Then it came to pass on the fourth day that they arose early in the morning, and he stood to depart ..."
b) 2 Ch 20:25b-26a "... and they were three days gathering the spoil because there was so much.
And on the fourth day they assembled in the Valley of Berachah, ..."
c) Josephus' Wars, V, VIII, 2 "Thus did they valiantly defend themselves for three days; but on the fourth day they could not support themselves against the vehement assaults of Titus"
d) Josephus' Ant., IX, I, 3 "He also gave his army leave to take the prey of the enemy's camp, and to spoil their dead bodies; and indeed so they did for three days together, till they were weary, so great was the number of the slain; and on the fourth day, all the people were gathered together unto a certain hollow place or valley, and blessed God for his power and assistance ..."
B) And the third day is the next one after two (NOT three) days (as the seventh day follows six days):
a) 2 Sa 1:1-2a YNG "And it cometh to pass, after the death of Saul, that David hath returned from smiting the Amalekite, and David dwelleth in Ziklag two days,
and it cometh to pass, on the third day, that lo, a man hath come in out of the camp from Saul ..."
b) Ex 16:26 "Six days you shall gather it, but on the seventh day, the Sabbath, there will be none."
c) Ex 20:11 "For in six days the LORD made the heavens and the earth, the sea, and all that is in them, and rested the seventh day."
d) Lev 23:3 "Six days shall work be done, but the seventh day [is] a Sabbath of solemn rest, a holy convocation. You shall do no work"
e) Jos 6:3-4 "... you shall go all around the city once. This you shall do six days. ... But the seventh day you shall march around the city seven times"
f) Josephus' Wars, II, XIX, 7 "There it was that Cestius staid two days, and was in great distress to know what he should do in these circumstances; but when on the third day he saw a still much greater number of enemies, and all the parts round about him full of Jews, he understood that his delay was to his own detriment ..."
g) Josephus' Wars, III, VIII, 1 "Thus he concealed himself two days; but on the third day, when they had taken a woman who had been with them, he was discovered."
h) Josephus' Wars, IV, VIII, 1 "... Cesarea to Antipatris, where he spent two days in settling the affairs of that city, and then, on the third day, he marched on, laying waste and burning all the neighboring villages."
i) Josephus' Ant., I, XIII, 2 "Now the two servants went along with him two days; but on the third day, as soon as he saw the mountain, he left those servants that were with him till then in the plain, and, having his son alone with him, he came to the mountain."
Besides, Mark 14:58, talking about Jesus' resurrection using the figure of the temple, in a verse that is not in the Mark 16 later addition, says: "within (dia) three days I will build another made without hands."
Yes, but when "Mark" had Jesus predicting his resurrection "after three days", he did not use 'dia'. Furthermore I do not see why Jesus' alleged claim about rebuilding the temple "within three days" has to match exactly the predicted duration of Jesus' death: "after three days".
Your position is that the Old Testament must be wrong and the king's statement must be an interpolation. But that's unnecessary. The king said to come back after three days, and they came on the third day. So what if they came to the king a day early? It doesn't make the people bad, so there is no embarrassing need to censor the text with an interpolation. And why do you think that it's an interpolation in the first place, or if it is, that it must be wrong? Because, it seems, you are relying apriori on your pre-existing belief that "after three days" must be meant by the ancient Jews in common speech strictly as we use it today, and not rather in some less precise colloquial way.
Actually, looking at:
After the king (Rehoboam) says "Depart for three days, then come back to me."
1 Ki 12:12 "So Jeroboam and all the people came to Rehoboam the third day,"
It makes sense that the people reassemble on the third day, in case the king would make his proclamation at the very beginning of the the fourth day (after three days), so they won't miss it (however, an interpolator, adding up the very awkward, erroneous and unnecessary "as the king had directed, saying, "Come back to me the third day.""
thought some correction was necessary).
To conclude, I do not think that 1 Ki 12:5, 12:12 (and parallel verses in 2 Chrononicles) can be used to indicate some ancient writers may have considered "after three days" compatible with "on the third day".