I'm asked to compare the homotopy type of two topological spaces, and I'm not sure what words to use.
For example, given $S^1$ and $S^1\lor S^1$, I know that the fundamental group of $S^1$ is a proper subgroup of the fundamental group $S^1\lor S^1$. I would like to say something like "The homotopy type of $S^1$ is smaller than the homotopy type of $S^1\lor S^1$".
Furthermore, I wouldn't like to rely on the subgroup relation, since any free group has as subgroup any other free group. In the previous example I could use the fact that the rank of $\mathbb{Z}$ is strictly smaller than the rank of $\mathbb{Z}*\mathbb{Z}$.
Question: Is there any common terminology for the comparison of homotopy types?
It doesn't really make much sense in general to compare homotopy types in more than the most basic language. For example, how do the homotopy types of $S^1$ and $S^\infty$ compare? Since $S^1$ includes into $S^\infty$, you might be tempted to say that $S^1$ is 'smaller' than $S^\infty$. But $S^\infty$ is weakly contractible, so contains a lot less homotopic information than even $S^1$. These spaces are simply homotopically different.
It does make sense, however, to say that a space $A$ is a retract, or deformation of another space $X$ (which is the case for your example). This is really the only case that you might be tempted to say that $A$ is 'smaller' than $X$, since all the homotopy and homology groups of $A$ appear as summands of those of $X$. Hence $X$ contains all the basic homotopic information of $A$, as well as possibly more. You still need to take care, though, since $A$ and $X$ may actually be homotopy equivalent.