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 (see p. xix), 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. Here, Jacquet credits M-W's proof to Colin de Verdière's "new and strikingly brief and elegant proof" (from the MR review) for $SL_2({\mathbb R})$ (extended here). I couldn't find any extensions of Colin de Verdière's argument to higher-rank cases, but that may be because it was done in Moeglin-Waldspurger.
Muller also has a proof in the rank-one case.
Wong gave a proof using integral equations.
Everyone listed credits Selberg with their ideas.