Syndet bars, melt and pour, and the composition test
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The other half of the exemption, and the half that is about what is in the bar rather than what you say about it.
The first condition, tested
The bulk of the non-volatile matter must be an alkali salt of fatty acids, and the detergent properties must be due to those compounds.
Two tests, and the second is the more searching: it asks what is actually doing the cleaning.
Syndet bars
A synthetic detergent bar looks like soap, is sold as soap and cleans by a different mechanism.
The detergent properties are due to synthetic surfactants rather than to alkali-fatty acid compounds, so the first condition fails and it is not exempt soap, whatever the wrapper says.
Which matters commercially, because syndet bars are popular for being milder and for working in hard water, and a maker moving into them has moved into the cosmetic regime at the same time.
Part-syndet bars are the awkward middle: the question is which component the detergency is due to, and a bar formulated so the syndets do the work is a syndet bar.
Melt and pour
A pre-made soap base, melted, colored, scented and poured.
The base is genuinely saponified, so it can meet the composition condition. What decides it is the base's composition, which is the supplier's to state and yours to check.
Many bases contain substantial glycerine, propylene glycol, sorbitol and sometimes syndets to make them clear and pourable, and at some point those stop being minor additions.
Ask the supplier directly what the base is, and keep the answer, because you are relying on their formulation for your own regulatory position.
Hybrid and superfatted bars
A heavily superfatted bar contains free oils that are not saponified, which is the whole point of superfatting.
At ordinary levels that does not disturb the bulk test. At very high levels the question becomes real.
And a bar with substantial non-soap additions, a large clay load, heavy butters stirred in at the end, is worth thinking about the same way.
The practical answer: an ordinary cold process bar at a normal superfat comfortably meets the composition condition, and the condition that fails is almost always the other one.