Some authors reserve the term for a special type of integral representation, namely those that involve an Eisenstein series.
Erich Hecke, and later Hans Maass, applied the same Mellin transform method to modular forms on the upper half-plane, after which Riemann's example can be seen as a special case.
The simultaneous combination of an unfolding together with global control over the analytic properties, is special and what makes the technique successful.
Hervé Jacquet and Robert Langlands later gave adelic integral representations for the standard, and tensor product L-functions that had been earlier obtained by Riemann, Hecke, Maass, Rankin, and Selberg.
Nowadays one has integral representations for a large constellation of automorphic L-functions, however with two frustrating caveats.