Just wanted to note that Meikyo also has a specific sense for "no thanks":
❸ 〔俗〕相手の勧誘などを遠回しに拒否する語。結構。

Understood. I use it that way too. I've identified a suitable example.

"When a waiter asks them “do you want some water?”, 
people say “daijobu desu” to mean “no thanks” or 
literally “I’m fine”. But if they ask “Could I pour some 
more water into your glass?” and people say “daijobu 
desu”, it can be easy to become confused as to what this 
“daijobu” indicates."
http://jpninfo.com/3702

Daijs 2 senses:
１ あぶなげがなく安心できるさま。強くてしっかりしているさま。
「地震にも大丈夫なようにできている」「食べても大丈夫ですか」
「病人はもう大丈夫だ」
２ まちがいがなくて確かなさま。「時間は大丈夫ですか」「大丈夫
だ、今度はうまくいくよ」
(gg5 is pretty much the same)

The thing is that to use "大丈夫" to decline an 
invitation etc. is considered incorrect/slang, so I 
don't think it should be tweaked into sense 1. The 
"alright/OK" in sense 1 is different from the 
"alright/OK" in sense 3. Sense 1: （具合の悪そうな人に）
「大丈夫ですか？」「ええ、大丈夫です」(from gg5)
It's an "OK/alright" as in, "not dying/safe/free from 
worry"
Sense 3: (from the Asahi article)
「“今晩、一杯飲みに行くか？”って誘ったら“大丈夫です”って言われ
たって言うんだ。“大丈夫”なら来られるはずだろう。いったい何が大
丈夫なんだ？」
_
Tried adjusting sense 1 a little to make it more obvious 
that it doesn't mean "OK/sure!" as in "sure, let's do 
it"

@@ -21,0 +22,4 @@
+<gloss>secure</gloss>
+<gloss>sound</gloss>
+<gloss>problem-free</gloss>
+<gloss>without fear</gloss>
@@ -26 +29,0 @@
-<gloss>sure</gloss>

I'm possibly missing something, but I really can't see the case for this third sense. I can't think of examples where it differs noticeably from sense 1. I checked the 78 Tanaka sentence for 大丈夫 (http://www.edrdg.org/cgi-bin/wwwjdic/wwwjdic?1Q�����_0__) and I can't see any there that would fit. GG5 only has the two - surely a little tweaking for sense 1 would do?