I'm not familiar enough with the proofs to say if they are more than superficially different, but here is something:
Moeglin-Waldspurger prove continuation crediting Jacquet (who cites Selberg), instead of Langlands. They say it is similar to that given by Efrat, in his treatment of the Hilbert-modular ($PSL_2$ over a totally real field) case.
Colin de Verdière gave a "new and strikingly brief and elegant proof" (from the MR review) for $SL_2({\mathbb R})$ (extended here). Muller also has a proof in the rank-one case. They both seem to be related to scattering theory (also see Lax-Phillips). It's worth noting that I can't find a treatment for general reductive groups using these methods.
Wong gave a proof using integral equations.
All the ideas seem to go back to Selberg.