"Cusp form" means "cuspidal automorphic form" by definition. So yes, $\theta(z;u)$ is an automorphic form. But it is not a form lying in a cuspidal automorphic representation, because it is not a Hecke eigenform. (Actually, this is slightly misleading, see the Added section for clarification.)

Cuspidal automorphic representations are irreducible subspaces of the relevant cuspidal automorphic $L^2$-space, so most cusp forms are linear combinations of forms coming from distinct (usually infinitely many) cuspidal automorphic representations. For non-cuspidal forms the picture is even more complicated: linear combination is replaced by an integral with respect to some spectral measure which includes various (Hecke-)Eisenstein series. The general theory is due to Langlands (spectral decomposition of $L^2$-automorphic forms). (Again, see the Added section for clarification. The point is that the notion of irreducibility depends on the group, and one can think of the group over the reals $\mathbb{R}$ but also over the rational adeles $\mathbb{A}_\mathbb{Q}$.)

**Added.** I realize now that a serious confusion can arise from considering (irreducible) cuspidal automorphic representations of $\mathrm{GL}_n(\mathbb{R})$ vs. considering those of $\mathrm{GL}_n(\mathbb{A}_\mathbb{Q})$. I like to consider the adelic group $\mathrm{GL}_n(\mathbb{A}_\mathbb{Q})$, because it has more structure and is more relevant for number theory. Then, a vector from a cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb{A}_\mathbb{Q})$ corresponds to a cusp form on $\mathrm{GL}_n(\mathbb{R})$, which is an eigenfunction of the local spherical Hecke algebra at all but finitely many places, and vice versa. One can refine this notion to talk about newforms and oldforms, by considering non-spherical Hecke algebras (but Hecke algebras corresponding to a level). In the end, the notion of "automorphic form" and "automorphic representation" varies greatly with the context. A good introduction is Borel-Jacquet: Automorphic forms and automorphic representations (Proc. Symp. Pure Math. 53 (1979), 189-202), which discusses both $\mathrm{GL}_n(\mathbb{R})$ and $\mathrm{GL}_n(\mathbb{A}_\mathbb{Q})$.