This is a good question; you've hit on a rarer use of elicit.
I think good dictionaries can bring us very close to this usage, but that a slight expansion or shift of the meaning is still in use. However, other users, such as Tᴚoɯɐuo, have given compelling reasons to accept it as standard, so you can make up your own mind.
My take is that it can be justified, but I wouldn't necessarily pick "elicit" first when to express what the author means here, which is "bring to light".
The Oxford English Dictionary lists three entries for elicit, which I'll quote in full and add emphasis:
1. trans. To draw forth (what is latent or potential) into sensible existence. Also fig.
2. To bring out, educe (principles, truths, etc.) from the data in which they are implied. Also, to extract, draw out (information) from a person by interrogation; sometimes with object clause introduced by that.
3. To draw forth, evoke (a response, manifestation, etc.) from a person.
What is clear is that meanings (1) and (3) can't mean "bring to light". They both mean causing something, provoking it, triggering it — in short, bringing it "into sensible [=noticeable] existence". Now, the approaches of the various countries have existed long before the G20 "occasion".
However, LawrenceC below interprets the "occasion" as the U.S.'s posturing on environmental change, and reasons that this is what has "elicited" (caused) the other countries' approach. To my mind, "elicit" implies too spontaneous a reaction to mean this, but it isn't unreasonable.
That last definition in the OED, (2) is interesting, since it does correspond to the notion of "bringing something to light". The countries' approaches or positions existed already, and the G20 summit is the occasion that revealed them or drew them into focus.
My reservation is that in the definition, this applies to "principles, truths, information". So it would have to be by metonymy that it would apply to "approach". What was literally elicited was the information about or realization of their approaches.
That said, there are precedents for this metonymy. The OED itself gives an example where this usage is clearly the one intended:
1875 B. Jowett in tr. Plato Dialogues (ed. 2) V. 88 The matter in dispute should be clearly elicited from the contending parties.
So I have to retract my comments above, even if I think it's not the most typical use of "elicit".
P.S. But as a general note, keep questioning newspaper and magazine articles. They often have to write and edit in haste, particularly when an article appears on the same day of an event (July 7 in this case was the first day of the summit), and I can say from experience that something always gets through no matter how many proofreaders or editors look at something. And even if the usage turns out to be justified, you learn something new. :)