Well "premium" and "regular" usually refer to the octane rating of the fuel. This is a measure of the ignition energy required to start the combustion reaction. Higher octane fuels require more energy to ignite. In higher compression engines the air/fuel charge heats up more during the compression stroke than in lower compression engines. If the fuel lights off at too low a temperature it will ignite too early and cause "detonation". This will cause the engine to run roughly and inefficiently and can cause damage to the engine.
Correct, but the point of high octane is you can run a higher compression ratio engine. The higher the compression, the greater the power for a given cylinder size. Of course, if you don't have a high-compression engine, then there's not much point in burning high octane fuel.
In the past lead was used as a detonation inhibitor.
No. In older engine designs, the valve guides in the cylinder heads required lubrication. Tetraethyl lead was added to the gasoline to provide this lubrication. Originally, there was a separate, more expensive grade of gas, known colloquially as "ethyl," that people would run a tank of every eight or ten tanks to provide this lubrication; then they eliminated this, putting it in all grades, in lesser amounts, because people would forget and damage their cylinder heads. But lead is known to be a poison, and also interferes with the action of the catalyst beads in a catalytic converter, so they had to make cars that could burn unleaded gasoline; to do this, they used bronze valve guides, because bronze is self-lubricating on steel valve stems. Today, you can't get heads that don't have bronze valve guides and require leaded gas; most even of the cars that originally required leaded gasoline have had their heads replaced with ones that can burn unleaded.
Contrary to what many people think the octane rating has nothing to do with the energy released during combustion. 93 octane is not "more powerful" than 87 octane. 93 octane fuel will release the same amount of energy as 87 octane fuel.
Correct. The point is you can burn it in a higher-compression engine, and get more power. It doesn't make more power without that extra compression, and the compression ratio is a design-time choice in an engine.
BTW the octane rating is an artificial scale based on octane and heptane. The seven carbon heptane hydrocarbon is easier to ignite and the eight carbon octane hydrocarbon is harder to ignite. 89 octane fuel behaves as though it it 89% octane and 11% heptane.
Correct, although it's not just heptane and octane, it's specific isomers of them: iso-octane and neo-heptane.