Though I don't know which book you are using, I shall assume that all graphs are simple (otherwise we may refer to them as multigraphs).
The case of cycles of length $2$ comes down to the difference between cycles and closed walks. Cycles are usually assumed to be simple, that is, not use any vertex or edge more than once (except that it starts and ends in the same vertex). For instance, we would not consider $C_4$ to have a cycle of length $6$, even though it does have a closed walk of length $6$. Therefore we also do not consider something like $\langle a,b,a\rangle$ to be a cycle.
The case of the single-vertex cycle $\langle a\rangle$ is a bit trickier. Couldn't we say the begin and end point of this cycle are the same, with no edges in between them? (After all, a simple graph is not allowed to have loops.) Well, yes, that would make it a cycle of length $0$. This is a degenerate case: it does not suit any practical use, and it defies the usual properties of cycles that we know (for instance: "the number of vertices on a cycle is the same as the number of edges"). Furthermore, it complicates various definitions (such as: "a forest is a graph without cycles", or "the girth of $G$ is the length of the shortest cycle in $G$"). For this reason the trivial cycle is often excluded.
Maybe somebody has a more compelling reason why a trivial cycle should be excluded, but I see it mostly as a matter of convention. In any case, rest assured: the definitions you encountered are common. In graph theory (and mathematics in general), sometimes definitions are modified to exclude degenerate cases or to prevent strange corner cases from turning up further down the line.