I see 新和英中辞典 splits "attempt" into its own sense, but I think that's a bit fine.

I think it's important to include "attempt" as probably the main gloss for this word.
-
I also think the definition needs glosses like "endeavor" and "initiative" that are less suggestive of the 
possibility of failure. Genius WaEi gives this sense as "endeavor 真剣な努力、尽力" 
A translator — not me this time :-) — was scolded by a client as follows:
「これは世界で初めての試みです。」を「 It is the first trial service of its kind. 」と訳す等、日本語のニュアンスを理解していないよう
に思える。
(This was in reference to a unique kind of business the client was starting up, but *not* on a trial basis, 
evidently. They expect the business to continue successfully.)
-
Here are some ALC examples, where I think "endeavor" is more appropriate than "experiment" or "trial":
1) インドネシアのインターネット・ブラックリストの試みは、東南アジア諸国に与える影響を考えると、監視していく必要がある。
Indonesia's plan to enforce an internet blacklist should be monitored because of its impact in the region.
[It's seen as a *plan*, not just an *attempt* or *experiment*.]
2) ～への具体的な試み
specific endeavor for
3) この試みにおいて
in this endeavor
4) このユニークな試みの真意は何なのか。
What is the significance of this unique venture?
5) この新しい試みのリーダーの一人は、ウォルト・ディズニー・パークスアンドリゾートの会長、ジェイ・ラスロ氏です。
One of the leaders of the new initiative is Jay Rasulo, who serves as chairman of Walt Disney Parks and 
Resorts.
6) この試みでは、米国を訪れた人々の間で米国に対する好意的な意見が42％上昇したことを示した調査を引き合いに出しています。
The initiative cites research that found a 42 percent increase in favorable opinion of the U.S. among those 
who have visited the country.
7) カナダやオーストラリア、米国では、同様の試みがすでに成功しています。
Similar efforts have already proven successful in Canada, Australia and the U.S.

Shogakukan has another sense "ordeal; temptation", as in sentences like the following one from ALC:
われらを試みにあわせず、
And lead us not into temptation.

@@ -18,0 +18,1 @@
+<gloss>attempt</gloss>
@@ -21,0 +22,7 @@
+<sense>
+<pos>&n;</pos>
+<gloss>endeavour (endeavor)</gloss>
+<gloss>effort</gloss>
+<gloss>venture</gloss>
+<gloss>initiative</gloss>
+</sense>