You don't need to explain to me what the theory of evolution is, or how it works. This question is purely about what exact meaning the word "natural selection" is ascribed to.
There seem to be multiple meanings that are given to the term "natural selection":
The process by which, in a single generation of species, the most fit individuals tend to survive. The "things that are being selected", are the individual organisms, and the "moment that selection took place", is the moment that these individuals reproduced or failed to reproduce, or alternatively, the moment that they died or survived.
The process by which, over many generations, the genotypes that result in highest fitness tend to to be selected for. The "things that are being selected" are the genes, and the "moment that selection took place" is not a single instance in time or a single generation, but a long time-span. In fact, I sometimes hear the idea that natural selection is already selecting for genes even if those particular genes don't exist in any organism. For example, if natural selection is selecting for "height", and there is a gene X that would increase height, then it is said that X is being selected for even if X does not yet exist.
Are both of these interpretations generally accepted?