[PeterWhittaker]

The basic distinction is this: Natural selection refers to one organism having superior reproduction prospects due to inherent physical factors (strength, speed, height, weight, ability to digest specific foods, ability to detect and respond appropriately to threats, etc.). Natural selection takes place without the awareness of the hosts involved: By the time their relative advantages or disadvantages are in evidence, they are long dead.

In other words, natural selection is about fit, that is, how well an organism "fits" or is suited to and environment. (That it what Darwin meant by fitness: Environmental suitability.)

Sexual selection refers to overt preferences in mating: A female bower bird chooses the male that builds the "best" bower (a completely useless artifice neither ever use), a human male prefers women with larger breasts (pretty much ditto), a peahen prefers the peacock with the more extravagant tail (ditto for sure, in fact, more extravagant tails are liabilities).

Examples are many. The term "runaway sexual selection" is used to refer to exaggerated traits that become prevalent and possibly dominant in specific, local populations (think of human characteristics you associate with certain groups: Sometime, somewhere, someone preferred those slightly larger eyes or slightly curvier hips and over time those became prevalent). These traits are generally completely irrelevant, with zero fitness value.

They exist because we think they look/smell good. Why did we come to prefer one over another? The same random variation that causes changes to fitness causes changes to fitness indicators, but those indicators are not necessarily honest signals.

[arketyp]

I am familiar with the gene-centered view of evolution proposed by Williams and popularized by Dawkins with the so-called selfish gene theory. Within this school of thought sexual selection by no means stands in contrast to natural selection and is not labeled an alternative to it. Instead, in fact, sexual selection is argued to be a direct consequence of natural selection as a phenomena you would expect with Darwinisitic reasoning when assuming the gene-centered view. As is made the case, the gene-centered view is the extension of Darwin's original argument, that is, Darwin's argument taken to its logical conclusion with the modern facts at hand, the DNA molecule not the least.

One of the central themes in Dawkins's The Extended Phenotype is the putting into question of the validity of traditional role of the organism in Darwinistic reasoning and why Darwin's original definition of fitness does not hold up. There is actually an entire segment on the many modern (back then) definitions of fitness and problems with each of them. I don't recall what the final conclusion is but it is certain it is not the clear-cut case you make it.

I suppose you have background in a field where your definition of fitness is more firmly established. For instance, I know biological mathematics has come a long way, and I'm not up to speed with what definitions they are using.