I'm not familiar enough with the proofs to say if they are more than superficially different, but here is something:

1. 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.

2. 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.

3. Wong gave a proof using integral equations.

All the ideas seem to go back to Selberg.

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.