Why is a heritability coefficient not an index of how "genetic" something is?
Rather than discussing what heritability is not through wordy sentences, let's just talk about what heritability is. There are two "types of heritability":


Heritability in the broad sense
Heritability in the narrow sense.


I will discuss a few concepts and slowly introduce the concept of heritability in both senses.

Phenotypic trait

The phenotype is the consequence of the genotype on the world. In brief, a phenotypic trait is any trait that an individual is made of! 

Quantitative trait

A quantitative trait is any trait that you can measure and ordinate, that is any trait that you can measure with numbers. For example, height is a quantitative trait as you can say that individual A is taller than individual B which is itself taller that individual C.

Variance of a quantitative trait

In a population, different individuals can have different values for a given phenotypic trait $x$. Because we are talking about quantitative traits we can calculate the variance of the trait in the population. Let's call this variance $V_P$ such as

$$V_P=\frac{1}{N}\sum_i (x_i - \bar x)^2$$

In the above equation, $x_i$ is the value of the phenotypic trait $x$ of individual $i$. $N$ is the population size (there are $N$ individuals in the population) and $\bar x$ is the average phenotypic trait $x$ in the population.

$$\bar x = \frac{1}{N}\sum_i x_i$$

What is causing phenotypic variance

Why would a population display any phenotypic variance? Why wouldn't we just look exactly the same? What explains these differences?

For some traits, we see very little variance. To consider the example the OP gave in the post, the number of arms in the human population shows very little variance. However, there is quite a bit of variance in terms of the number of IQ, in terms of height or of weight.

There are two (main) sources of variance that are underlying this phenotypic variance. The first one is the genetic variance and the second one is the environmental variance. We will call the genetic variance $V_G$ and the environment variance $V_E$.

If in a population, people vary a lot in terms of how many hamburgers they eat, then there is a non-negligible $V_E$ underlying the phenotypic variance $V_P$ for weight. If in a population, there is a lot of variation of genes affecting weight, then there is a non-negligible $V_G$ underlying the phenotypic variance $V_P$ for weight.

By the way, a gene (or another non-coding sequence) that is polymorphic (i.e. has more than 1 allele in the population) and which explains some of the variance in the phenotypic quantitative trait is called a Quantitative Trait Locus (QTL). A locus is a sequence (of any length) on the genome. 

Math reminder

Variances of uncorrelated variables can simply be added! For simplicity, we will assume for the moment that we are considering uncorrelated variables. As a consequence, we can express the phenotypic variance $V_P$ as a sum of the phenotypic variance that is due to environmental variance $V_E$ and the phenotypic variance that is due to genetic variance $V_G$

$$V_P=V_E+V_G$$

This equation is slightly simplified and this will affect the below calculations. See the section Other sources of phenotypic variance for more info.

We can now talk about heritability!

Heritability in the broad sense

Heritability in the broad sense $h_B$ is defined as the fraction of phenotypic variance $V_P$ that is explained by genetic variance $V_G$. In the equation, it gives:

$$h_B=\frac{V_G}{V_P} = \frac{V_G}{V_E+V_G}$$

Heritability in the narrow sense

Heritability in the narrow sense $h_N$ makes one further trick. We have to consider that the genetic variance $V_G$ that is underlying the phenotypic variance can itself be decomposed into a sum of variances. The variances that we like to consider the additive genetic variance $V_{G,A}$ and the dominance genetic variance $V_{G,D}$.

The additive genetic variance is the genetic variance that is due to additive interaction between alleles. The dominance of genetic variance is due to non-additive interactions between allele.

We can now define the heritability in the narrow sense $h_N$ as the is defined as the fraction of phenotypic variance $V_P$ that is explained by the additive genetic variance $V_{G,A}$. In the equation, it gives:

$$h_N=\frac{V_{G,A}}{V_P} = \frac{V_{G,A}}{V_E+V_G} = \frac{V_{G,A}}{V_E+V_{G,A}+V_{G,D}}$$

In the special case, when all the genetic variance $V_G$ is exclusively done through additive interactions, then $V_{G,D} = 0$ and $V_{G,A}=V_G$ and therefore $h_N=h_B$

Interpretation of the heritability

If all of the phenotypic variance is due to genetic causes (and regardless of whether there is a lot or a little variance), then $h_B=1$. If all of the phenotypic variance is due to environmental variance, then $h_B=0$.

So what does a $h_B=0.3$ means?

It means that 30% of the phenotypic variance is explained by genetic variance and that 70% of the phenotypic variance is due to environmental variance.

So, what if there is no phenotypic variance in the population? if $V_P=0$, then the heritability is undefined (as dividing by zero is undefined). However, in general, we tend to think that there is always a tiny bit of environmental variance and most people would just go on saying that heritability is 0 when $V_P=0$.

What will affect the heritability?

A measure of heritability is true for one population, in one environment.

If you change the population, add a few mutations for example, you might well create a polymorphic locus that is causing some phenotypic variance. If you put the same population in another environment, you could suddenly have more or less phenotypic variation due to environmental variance. Typically, if you measure heritability in the lab in a controlled and constant environment, then you will likely overestimate the heritability (as you underestimate $V_e$) compared to the same population that is living in a very heterogeneous environment.

What heritability is not!

If a trait has low heritability, it does NOT mean that it is (or is not) an adaptation. It only means that there is no genetic variance that explains the phenotypic variance.

Why do we care about heritability?

If there is no genetic variance for a trait, it means that the only way this trait can change through time is by changing the environment (or by creating a non-zero genetic variance through mutations). If there is a non-zero genetic variance and if there is a difference in fitness between individuals having different trait value then, the trait is under natural selection.

The most commonly used index of heritability in the heritability in the narrow sense $h_N. $Why do we care about $h_N$?

Let $\bar x_t$ be the mean phenotypic value of the trait $x$ at time $t$. One generation later, that is at time $t+1$, the mean phenotypic value is $\bar x_{t+1}$. Let's define the response of selection $R$ as the expected difference between these two quantities, that $R=E[\bar x_{t+1} - \bar x_t]$. Let's define the strength of selection $S$ and the heritability in the narrow sense $h_N$, then 

$$R=h_N \cdot S$$

As a consequence knowing $h_N$ allows us to predict the effect of selection on a given trait.

This equation is called the breeder's equation (see this post about its interpretation).

Other sources of phenotypic variance

Saying $V_P=V_G+V_E$ is a little too simplistic. In reality, there are other sources of phenotypic variation such as variance due to epigenetic changes $V_I$ and variance due to developmental noise $V_{DN}$ for example. It is also sometimes very important to consider the covariance between any pair of such variance. So, the equation would more correct if stated as

$$V_P = V_G + V_E + V_I + V_{DN} + COV(V_G, V_E) + COV(V_G, V_I) + COV(V_G, V_{DN}) + COV(V_E, V_I) + COV(V_E, V_{DN}) + COV(V_I, V_{DN})$$

Note that everyone is free to further decompose any of the above variance into a sum of variances as we did above for the genetic variance. For example, the environmental variance $V_E$ could be decomposed into the sum of the phenotypic variance due to variance in temperature $V_T$ and the phenotypic variance due to variance in precipitation $V_{\text{precipitation}}$ assuming the other types of environmental variances are negligible.