When does the NEC actually require limiting a 15A branch circuit to 12 amps, and when is loading it to the full 15A OK?

Answer is:

The 12-amp limit only applies in specific situations, not by default.

The 80% limitation in 210.23(B)(1) applies only when a branch circuit also supplies lighting outlets, lighting units, or other utilization equipment. In that case, any one cord-and-plug-connected load not fastened in place is limited to 80% of the circuit rating.

If the branch circuit does not supply other outlets or equipment, then that rule does not apply and the general rule governs—the load simply cannot exceed the branch-circuit rating.

Separately, if a load is expected to operate at maximum current for three hours or more, it is considered a continuous load and must also be limited to 80%. If it is not continuous, that restriction does not apply.

Bottom line: A 15-amp circuit may be loaded to 15 amps when properly applied and non-continuous; the 12-amp limit is conditional, not automatic.