I think few people would argue claims that Langlands' SLN 544 was written "in a different time" (e.g., pre-rigidity-theorems), and never really edited carefully (although the retyping is a relief), nor that Moeglin-Waldspurger aim for maximum generality/encyclopedic-ness, ... Selberg's sketches were of an even more different time. My colleague D. Hejhal confirms my speculation that Selberg did not think about cuspidal-data Eisenstein series, insofar as there was no mention of any such thing in his papers.
As I have noted elsewhere, Bernstein's ideas about meromorphic continuation of Eisenstein series are not well-documented, or, perhaps, not documented. Others may disagree, but I think that the gossip about "non-cuspidal-data Eisenstein series" being treated in any sane fashion is unreasonable...
To finally respond to the question: I think there does not currently exist any friendly, yet actually proof-y, expository treatment of any sort of general case of meromorphic continuation of Eisenstein series. (More on this below.)
At the same time, many people know very well "what is expected of" Eisenstein series, what role they play in spectral theory, and so on.
Indeed, this is a poster-child for the fact that frequently the function of an idea/thing is much simpler than its construction/justification. (Precedents: the real numbers, ...)
F. Shahidi's relatively recent book with AMS does show the general function of Eisenstein series in a sort of Langlands-programme-y setting, and in that regard is an excellent model for "using Eisenstein series". Indeed, most people should look "forward" in this way rather than worrying about whether or not some foundational thing was "really proven"...
The more direct questions about meromorphic continuation of Eisenstein series... are harder to usefully answer, unfortunately. If it's any comfort, the perception that these things are ... "not clear"... is easily arguably correct. As in some other MO comments/answers of mine, by this year I think that the most believable/persuasive argument for meromorphic continuation (of Eis with cuspidal data... which, the point is, enter in an $L^2$ spectral decomposition with respect to Casimir) is a refined form of the Colin-de-Verdiere/Lax-Phillips/Faddeev-Pavlov "discretization" arguments. That is, disappointing to me as well, the Selberg and/or Selberg/Bernstein arguments, while excellent high-level technical heuristics, have stunningly high hidden technical costs. Probably Selberg did not think in such terms, and it is pointless for me to conjecture what Bernstein thinks is trivial or not. :)
While I myself do have some plans to write up examples about Eisenstein series and spectral decompositions... based substantially on my on-line notes and "vignettes", I think the best guide is thinking consciously of the distinction between "definition/proof-of-foundational-properties" and "role in ...". That is, the assertion of the $L^2$ spectral decomposition (as representations? as Casimir-eigenspaces? ...) can be formulated without knowing the details of the proofs of meromorphic continuation ... as interesting as they might be. :)
EDIT: it may be worthwhile to mention that I did eventually fairly-carefully write up a family of examples of how to meromorphically continue Eisenstein series... among other things. There is a link at http://www.math.umn.edu/~garrett/m/v/ to a (legal!) PDF of my book "Modern Analysis of Automorphic forms, by Example", published with Cambridge Univ Press. Among other goals, one was to give legitimate, genuine analytical grounding.