Dominance coefficient
Thank you for asking such a well-thought out question, I really enjoyed how hard this made me think about the concept. I think there might be an issue here that's quite common to biology, in my experience- poor definition of terminologies.
I think the core question can be summed up by your comment:

I thought dominance was only related to which phenotype will show. If fitnesses of $A_{11}$ and $A_{22}$ are equal, doesn't it mean just that $A_{11}$ and $A_{22}$ have the same chance at surviving?

In my thinking, the definition of dominant could mean two things:

The expression of a particular characteristic
The fitness associated with the expression of a characteristic

Of course, the second definition depends entirely on the first- if the phenotype of the heterozygote ($A_{12}$) is the same as the homozygote ($A_{11}$ or $A_{22}$), then their fitness must be the same- evolution acts on phenotypes, not genotypes.
Let me break this down using Gillespie's tables. For a given value of $s$ (between 0 and 1), we can imagine the following scenarios for the value of $h$:




$h$
Phenotype
$A_1A_2$ fitness
Dominance scenario




h=0
$A_1A_2$ phenotype = $A_1A_1$ phenotype
$1-s\times 0 = 1 = A_1A_1 fitness$
$A_1$ is dominant allele


h=1
$A_1A_2$ phenotype = $A_2A_2$ phenotype
$1-s\times 1 = 1 - s = A_2A_2 fitness$
$A_2$ is dominant allele




This is assuming the phenotype for $A_1A_2$ is the same as the phenotype of the homozygote, in each scenario. It may be possible that the phenotypes are different, but the fitness is the same- what we are actually measuring with $h$ and $s$ are fitness, not phenotypes. But if the phenotypes are the same, the fitnesses must also be the same- natural selection works on phenotypes, not genotypes.
Where the break between fitness and phenotype becomes complete is when $h$ is a value other than 0 or 1 (not complete dominance).




$h$
Dominance Type
Scenario




0&lt;h&lt;1
Incomplete dominance
The heterozygote ($A_1A_2$) more fit than $A_2A_2$ and less fit than $A_1A_1$


h&lt;0
Overdominance
The heterozygote ($A_1A_2$) more fit than homozygotes ($A_1A_1$ and $A_2A_2$)


h&gt;1
Underdominance
The heterozygote ($A_1A_2$) less fit than  homozygotes ($A_1A_1$ and $A_2A_2$)




In each of these cases, the phenotype of the heterozygote is something different than either of the homozygotes (hence the term incomplete dominance). What the value $h$ is measuring here is not the difference in phenotypes, it's the effect it has on fitness- which might be different in different contexts.
Let's think about the classic example of overdominance, sickle cell disease. In terms of fitness, individuals with one copy of the sickle cell allele have the sickle cell trait. The sickle cell trait offers resistance to malaria, and therefore higher fitness in regions with endemic malaria. An individual with two copies of the sickle cell allele are less fit because they have sickle cell disease. An individual with no copies of the sickle cell allele is less fit because they don't have resistance to malaria infection.
But this measure of dominance, where we define dominance as the fitness associated with a the expression of a characteristic, is context-dependent- individuals with the sickle cell trait are prone to other diseases, and in the absence of endemic malaria it might be they have lower fitness than an individual with no copies of the sickle cell allele ($0&lt;h&lt;1$).
If we think about dominance as simply the expression of a characteristic, we would actually describe the sickle cell heterozygote as co-dominant- when you look at the blood of individuals with sickle cell trait (heterozygotes), they have a mix of normal-shaped red blood cells and sickle-shaped red blood cells.
So to conclude:

Dominance sometimes refers to the expression of a trait in relation to an allele, and sometimes refers to the fitness associated with the expression of a trait in relation to an allele.
Differences in fitness imply differences in phenotype (because natural selection acts on phenotypes, not genotypes)
Differences in phenotype do not necessarily imply differences in fitness


I wrote also some notes on the following, but as they're not directly relevant to the question (but might be useful for Googlers later on) I will put them here below the cut:
Table of relative fitness measures with dominance coefficient $h$ and selection coefficient $s$




Genotype
$A_1A_1$
$A_1A_2$
$A_2A2$




Relative fitness
$1$
$1-hs$
$1-s$





The choice of alleles assigned to $A_1$ and $A_2$ are arbitrary- in reality these are As, Ts, Cs, and Gs. By convention the first allele is the dominant one, but if you're not sure which is dominant before you test it may be the second!
Similarly, by convention we assume the individual with $A_2A_2$ is not more fit than individual with $A_1A_1$ (in other words, $0&lt;s&lt;1$). But you might have it backwards!