What is the definition of "Natural Selection"?
Official definition


  Is there an official definition of natural selection that is adopted by biologists nowadays? and what is that definition exactly?


I don't think there is such a concept as an "official definition" of any concept in science. There are common definitions though.

The definitions you cite

Let's go through your three definitions


  Natural selection is the differential survival and reproduction of individuals due to differences in phenotype.


This definition lacks the idea that this fitness differential must be caused, in part at least, by genetic variation.


  Natural selection, a process that results in the adaptation of an organism to its environment by means of selectively reproducing changes in its genotype or genetic constitution.


Part of the phrasing is about the consequence of the process rather than the process itself. The part that describes the process itself is "selectively reproducing changes in its genotype, or genetic constitution", which is a rather unusual phrasing and I would argue is slightly vague.

Note that while most authors refer to natural selection as the fitness differential, it is not impossible to refer to natural selection as the resulting change in allele frequency. I have never seen cases where this semantic difference matters though.


  Natural selection, then, can be defined as the differential reproduction of alternative hereditary variants, determined by the fact that some variants increase the likelihood that the organisms having them will survive and reproduce more successfully than will organisms carrying alternative variants.


I have never encountered the expression "alternative hereditary variants". I think the author made it up. It sounds like the author tried to avoid the term "allele" by replacing it with "alternative hereditary variants". The author also seems to have tried to avoid the term "fitness" by replacing it with "likelihood that the organisms having them will survive and reproduce more successfully".

But overall the definition sounds correct to me. There is an important difference between the first and the third definition is that in the first definition, it emphasizes the fact that the genetic elements that explain fitness variance are doing so via its effect on the phenotype. The third definition just ignores phenotype. As such, I would consider the first definition more complete (and more compact).


  Natural Selection is a mechanism of evolution that leads to non-random spread of genes due to the effect that genes have on reproductive success


Here, the definition is again very much based on the consequence of the process rather than the process itself. Only "due to the effect that genes have on reproductive success" refers to the actual process of natural selection.

I don't really like the term "random" here as the concept of randomness is somewhat unpleasant to define outside of a well-defined model. I don't really like the term "genes" either because selection doesn't only act on genes (but can act at any locus) and because the term "allele" would be more appropriate even if the phrasing would still remain unclear.

So in short, shame on us, the StackExchange definition is pretty bad!


  The preservation of favourable variations and the rejection of injurious variations


I shall not talk here about how different is Darwin's theory of evolution from today's theory of evolution (which is called the "modern evolutionary synthesis"). I shall not talk either whether this single short sentence is a good summary of Darwin's representation of natural selection (see @Luaan comments for more information).

This definition is short and a bit vague. One would want to think of this definition as applying to a genetic variant, without regard to the phenotype, however, Darwin did not know anything about DNA. 

My favourite definitions

Lewontin's recipe

A very simple way to think of natural selection is to consider Lewontin recipe. The Lewontin recipe says that natural selection occurs whenever:


Individuals in a population vary in terms of a given trait
This trait has a non-zero (additive) heritability. See this post for a definition of heritability.
The fitness varies (not necessarily linearly) as the trait varies


Single sentence definitions

If you prefer a shorter and simpler version I would personally suggest


  Selection is the fitness differential among genotypes in the population.


This definition lacks the concept of phenotype but talking about the fact that an individual fitness is dependent on its phenotype (among other things) is, IMO, not required to mention in the definition of what natural selection is. Not that I also removed the term "natural". I don't quite like it because it leads people to think that natural and artificial selection could be fundamentally different. Same applies to sexual selection, IMO, but I shall not go down this slope now!

In a very similar post, I wrote the definition


  Selection is a fitness differential associated to a genetic variance among individuals in a population.


I also like this definition because it highlights that selection acts on populations (and not on individuals or on species). 

Mathematical modelling

You might want also to consider as a definition the mathematical modelling of the process. In absence of genetic drift (assuming infinite population size), assuming a bi-allelic locus with alleles a and A, one can model (haploid) selection as

$$p_{t+1} = \frac{p_t W_a}{p_t W_a + (1-p_t) W_A}$$

, where $p_t$ is the frequency of the allele a at time $t$. $W_a$ and $W_A$ are the fitness of the genotype with alleles a and A, respectively. Of course, you can extend this model to any kind of ploidy and any number of alleles. You can also extend this model to include genetic drift (typically done via the use of diffusion equations or via branching patterns).

For more information about these mathematical modelling, please have a look at an intro book in evolutionary genetics (see here for book recommendation).