Poi?
 

A1 is dribbling in his/her backcourt and is fouled by B1. A1 is awarded a 1-and-1. Two situations now:

1) A1 makes the first, and then the table informs the officials that Team B only has 6 team fouls in the half.

or

2) A1 makes the first free throw but misses the second, which is rebounded by B2. The table informs the officials that Team B only has 6 team fouls in the half.

In both cases, we obviously take the point off the board. My question is where is the ball put in play to resume after the correction.

Is it on the endline or at the spot closest to where B1 committed the foul? Is the answer different in the two cases?

From the NCAA casebook
 

A.R. 22: Before the bonus rule is in effect, B1 fouls A1. The official errs by awarding A1 a one-and-one attempt.
(1) A1 makes the first free-throw try, and the error is then discovered; or
(2) A1 is successful in both free throws and then the official detects the error; or
(3) A1 misses the front end of the one-and-one and the game clock starts, at which time the official detects the error; or
(4) A1 is successful in the first bonus attempt but misses the second free throw and, as a result, the game clock starts and B1 scores a field goal. In each of the four situations, the error shall be called to the attention of the official before or during the first dead ball after the game clock has been properly started.
RULING: Each of the four situations is a correctable error.
(1), (2) and (3) Team A shall be awarded the ball at a designated spot nearest to where the foul occurred. The free throw(s) and activity during it, other than any technical foul or a flagrant 1 or 2 personal foul, shall be canceled. Any points scored, time consumed and additional activity that may occur before the recognition of the error, shall not be nullified.
(4) Team A shall be awarded the ball out of bounds at Team B’s end line, which was the point of interruption, to correct the error. The free throw(s) and activity during it, other than any technical foul or a flagrant 1 or 2 personal foul, shall be canceled. Any points scored, time consumed and additional activity that may occur before the recognition of the error, shall not be nullified.
(Rule 2-12-1.b, 2-12.5 and 2-12.2)

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Your second situation is not covered by this. In this case, you'd give the ball back to B at the designated spot nearest to where B2 was when the whistle was blown.

2008-2009 Case Book 2.10.1 SITUATION G: A1 is fouled prior to the bonus, but erroneously A1 is
awarded a one-and-one. The error is discovered: (a) after A1 makes the first
throw; (b) after A1 makes both; (c) after B1 has rebounded the miss on the sec-
ond free throw; (d) after B1 has the ball for a throw-in after both attempts are successful; or (e) after B2 has control of the throw-in from B1 after A1's two successful throws. RULING: In (a) and (b), the successful free throw(s) is canceled, and A1 is given the ball for a throw-in at the spot out of bounds nearest to where B1 fouled originally. In (c), (d) and (e), the successful free throw(s) are canceled and play continues with a throw-in by B as B had the ball when the game was interrupted for correction.
I don't want to be anywhere near coach A when I hand B1 the ball

Hopefully in those cases coach A is the home team, so you can tell him it was his table that screwed up...

Is anyone else bothered by the fact that the parts in red are not actually the point where we interrupted the game to correct the error?

True, but I think that I can justify it BY RULE for part a of the NFHS Case Book play, which this year is 2.10.1 Sit H.

4-36-2a doesn't apply as during the dead ball following the erroneous FT neither team was in control. So go to 4-36-2b. If we cancel the FT, one can certainly contend that Team A is entitled to a throw-in from the foul. Therefore, this rule would apply and the throw-in spot would be from the location that it should have occurred.

The problem is that part b doesn't match up with the text of the rule. After two successful FTs, Team B would now be entitled to a throw-in along the endline. That is the POI. It is Team B's ball.

I'm not sure that b doesn't match up. We have a three step test for restarting from a POI. First give it to the team that had control. Provided that B hasn't grabbed the ball yet, no team is in control.

Second, give it to the team entitled to a free throw or throw-in. A is entitled to a throw-in due to a common foul. B was entitled to a throw-in due to a made basket but that basket has been cancelled. Why then would B still be entitled to the throw-in?

We don't reach the third step.