Can domestic breeding be considered evolution?
Yes, selective breeding results in evolution.

Definitions of evolution

I don't understand why you say that for 3 of the definitions you found, selective breeding would not be considered as evolution. To me, all of these definitions match with the idea that selective breeding results in evolution. If you think otherwise, can you please explain why?

There are a number of related definition out there, some of them being more clear and straight forward than others. The simplest definition is probably the classical population genetics definition which says


  Evolution is a change in allele frequency over time.


See wikipedia > allele. You might want to have a look at How to define evolution? too.

Under this definition, any event of death or birth in a population is evolution as it affects allele frequency. In the human species for example, that means an evolution happens about 254 times per second (computed from these numbers from ecology.com)!

Specific statements

All these statements are stupid and none-sense! The easiest is probably for you to just have a look at an intro course to evolutionary biology (such as Understanding Evolution by UC Berkeley for example) and you will be able to refute those statements yourself.

Let's go through these statements...


  
  There is no natural selection
  


First, evolution is much more than just natural selection. There is genetic drift, mutations, gene flow and plenty of interacting processes.

Second, it is silly to separate natural selection from artificial selection (aka. selective breeding). The only difference between these two is that in artificial selection someone is willingly affecting the environment of the individual to affect selection pressures. This is really it.

Note, btw, you never totally get rid of natural selection. Under selective breeding you may affect the environment but some things will remain the same. For example, a mutation causing the cell membranes to be extremely unstable will not be selected for whether under an artificial setting or a natural one.


  
  It doesn't direct towards 'fitness'
  


The claim is unclear. What is "it" and assuming it refers to a population, then what does it mean a population is directing toward "fitness"?

Fitness is a function of survival and fecundity. Each genotype is associated to a particular fitness in a specific environment. Ignoring stochastic processes such as genetic drift, the mean fitness of a population is increasing from one generation to the other by the amount of the additive genetic variance. This is true for (artificial or natural) selection.

The only thing to consider is that the environments under which this happen is different. Under selective breeding, the population evolves to a higher fitness for the specific environmental conditions set by the farmer (or who ever who selectively breed). It is possible (and quite common in fact) that genotypes associated with a high fitness under one environment has a low fitness under another environment.


  
  It does not lead to new species
  


In short, 1) it does lead to new species 2) the concept of species is often meaningless as poorly defined 3) evolution > speciation. In more details, below..


It does lead to new species. Different lineages of cabbage are considered different species. Cows and ox are different species. Pigs and boars are often considered different species. While wolves and dogs are considered same species, some lineages within this species (such as a Chihuahua and a Great Dane) are, I think, reproductively isolated. You might also want to have a look at the post Have we ever observed two drosophila lineages that evolved reproductive isolation in labs?
The question of whether selective breeding lead to speciation or not does not matter much on the question of whether it leads to evolution. Speciation is one outcome of evolution but is definitely not the same as evolution. Evolution does not need to lead to speciation. For example, evolution of the lactase gene in humans (see this post) did not lead to any speciation. It is still an evolutionary process.
The concept of species is mainly arbitrary. If you want to understand the concept of species, have a look at the post How could humans have interbred with Neanderthals if we're a different species?.



  
  It decreases, rather than increases, the size of the gene pool (is this actually true?)
  


The term "size of the gene pool" is very undefined. I suppose, it just means genetic diversity, aka. expected heterozygosity.

All act of selection, whether artificial or natural, decreases genetic diversity. This is correct. Mutations is the fundamental process increasing this genetic diversity and selection selects the variants associate with the highest fitness. 

Because, the selection pressures imposed under selective breeding are often extreme, the decrease of genetic diversity via selection is often faster in selective breeding than in nature.

Note however, that if we split a lineage in two and select the two populations under different optimas, then the total genetic diversity will increased (the within-population genetic diversity will still decrease). As such, via selective breeding we created a gigantic diversity of dogs and a fantastic diversity of cabbage but each cabbage lineage (e.g. brussel sprouts) and each dog lineage (cocker spaniel) have very low genetic diversity.

What the claim is suggesting is that because selection is often stronger (and therefore change in allele frequencies is faster) under selective breeding than in natural environments, we should not consider artificial selection as resulting in evolution? How silly is that!