Q:

What is the geometric motivation for quotienting out boundaries in homology?

I know that being contractible is strictly stronger than being homotopically trivial, see e.g. this question. Now, trying to understand p. 100 of Hatcher's Algebraic Topology, the intuitive reason we factor out $n-$cycles which enclose $n+1$-cycles might be because they are homotopically trivial.
The first example on p.100 suggests that $1$-cycles that are the boundaries of $2-$cycles can be contracted to a point, and thus are not only homotopically trivial but also contractible.
How generally applicable is this way of thinking? Is "we factor out $n$-cycles that enclose $n+1$-cycles to account for the fact that they are contractible" valid geometric intuition for the definition of homology groups? (Or at least for cellular or singular homology?)
The idea seems to be along the lines that "boundaries of (n+1)-cycles correspond geometrically to the absence of holes, and thus we don't want to count them" and "n-dimensional cycles that are not boundaries of (n+1)-cycles enclose n-dimensional holes". Is this way of thinking correct?
If not contractibility, is there any formal way we can express the fact that "boundaries of (n+1)-cycles correspond to the absence of holes", besides per fiat?

The graph $X_1$: 
Let us now enlarge the preceding graph $X_1$ by attaching a 2-cell $A$ along the cycle $a-b$, producing a 2-dimensional cell-complex $X_2$. If we think of the 2-cell $A$ as being oriented clockwise, then we can regard its boundary as the cycle $a-b$. This cycle is now homotopically trivial since we can contract it to a point by sliding over $A$. In other words, it no longer encloses a hole in $X_2$. This suggests that we form a quotient of the group of cycles in the preceding examples by factoring out the subgroup generated by $a-b$. In this quotient the cycles $a-c$ and $b-c$, for example, become equivalent, consistent with the fact that they are homotopic in $X_2$.
$X_2$: 
Algebraically we can define now a pair of homomorphisms $C_2 \overset{\partial_2}{\to} C_1 \overset{\partial_1}{\rightarrow} C_0$ where $C_2$ is the infinite cyclic group generated by $A$ and $\partial_2(A)=a-b$. The map $\partial_1$ is the boundary homomorphism in the previous example. The quotient group we are interested in is Ker$\partial_1$/Im $\partial_2$, the kernel of $\partial_1$ module the image of $\partial_2$, or in other words, the 1-dimensional cycles modulo those that are boundaries [of 2-dimensional cycles], the multiples of $a-b$. The quotient group is the homology group $H_1(X_2)$... In the present example $H_1(X_2)$ is free abelian on two generators, $b-c$ and $c-d$, expressing the geometric fact that by filling in the 2-cell $A$ we have reduced the number of 'holes' in our space from three to two.

My question being: how do we express the motivation for modding/quotienting/factoring out the boundaries in topological terms? In geometric terms?

A:

This is just one example. In this example, there is only one 2-cell, and its boundary cycle $a-b$ is what is contractible. 
To continue the quote, "...we form a quotient ... by factoring out the subgroup generated by $a-b$". 
In other examples, you will have a larger number of generating elements that are being quotiented out. Those generating elements will generate, by the laws of algebra, an entire subgroup. That subgroup is called the "boundary subgroup". In the basic construction of homology, perhaps the generating elements of the boundary subgroup are contractible in some sense. But the group of cycles is an abelian group, and the formal algebra of abelian groups is forcing many other cycles to be quotiented out. In fact, the boundary group is nothing more nor less than the group of linear combination of the generating elements. So while certain generating elements have been singled out for special behavior, e.g. for being "contractible" in some sense, you can not make any deductions about the "contractibility" of a general element of the boundary group.
ADDED: Here's a few more thoughts, regarding what is intuitive in homology. Homology is abelian, unlike the fundamental group. This makes it computable and comparable in great generality, unlike the fundamental group. This whole business of "cycles" and "boundaries" is kind of what is forced upon you when you start with the fundamental group properties of "paths" and "homotopies" and attempt to abelianize them. In fact, one test of homology theory is that if you take the fundamental group, and abelianize it (a purely algebraic operation for converting arbitrary groups into abelian groups), the result turns out to be the 1st homology group. 

