Is this for you at all
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What this covers
Travel bottles for shampoo, lotion, sunscreen, mouthwash and perfume, plus the generic decant set people buy for a wash bag. Seven leaves, and for once the single archetype is close to right: they are all thin liquids in small bottles.
The hazmat rule
Non-radioactive medicinal and toilet articles for personal use, including aerosols, are excepted from the hazardous materials subchapter in carry-on and checked baggage, provided no single container exceeds 500 mL and each person carries no more than 2 liters in aggregate.
The security rule
TSA restricts carry-on liquids to a much smaller container as a security matter rather than a hazardous materials one. It is set by TSA, it changes without a rulemaking, and it is the tighter of the two, so it is the one that decides a carry-on bottle.
Which is why both matter
A 500 mL bottle is fine in checked baggage under the hazmat exception and will not pass carry-on screening. A product sold as a travel size has to answer the stricter rule to be useful, and the looser one to be lawful.
And then the physics
The archetype says it in one line: leakproof under pressure change is the whole requirement. A bottle sealed at ground level is at a higher internal pressure than the cabin once the aircraft climbs, and it pushes outward through the weakest part of the closure.
Two of the seven bring their own rules
A perfume decant is a Class 3 flammable liquid and does not stop being one by being small. A travel mouthwash can sit under the 3 g ethanol threshold that the full size clears, which is a real advantage worth knowing.
And one of them should not be decanted at all
Contact lens solution is required to be sterile, and pouring it into a bottle from a drawer destroys exactly that. The what-goes-wrong section says so plainly rather than selling a bottle for it.