Clarification please re: BR advancing to second?

[RonlyBonly]

Looking for some clarification please on a BR advancing to 2B after "overrunning 1B" or "making the turn" at 1B. The rules I am reading are ASA 8-7-B, 8-7-H, and POE 1-A-4.

Please tell me if my understanding is correct or if not, where it is screwed up. My understanding has been:

1) BR overruns 1B, turns right into foul territory to return to 1B is not liable to be put out while in foul territory. (?)
2) BR overruns 1B, turns left into fair territory to return to 1B is liable to be put out by being tagged while in fair territory and not in contact with the base. (?)

8-7-B seems to indicate BR is out in situation #2, or does 8-7-H mean that the BR has to be attempting to advance to 2B to be liable to be tagged out? IOW if the BR overruns 1B and is clearly just returning to 1B without any attempt to advance but is doing so inside the line in fair territory, is she liable to be tagged out as she returns to 1B?

3) In POE 1-A-4, "making the turn" refers to turning left into fair territory, correct?

POE 1-A-4 seems to put "attempting to advance to second" in a special category with "leaving a base on a caught fly ball before the ball is first touched" rather than it just being a runner not in contact with the base. I understand how the runner can be put out by tagging the runner while she is off the base (for whatever reason--whether she left a base early, is advancing to second, or just went to look at a bug) but why is "advancing to second" an appeal play rather than a routine tag out of a player off a base like any other?

POE 1-B and 1-C do not mention any live-ball or dead-ball appeal on "attempting to advance to second" other than tagging the runner so I take that to mean there is none.

Thank you.