Yoichi, I swear, there should be a term (perhaps "Yoichi-question-ism!") for (A) when in English we further play on a common figure of speech.
But then, more critically, there should be a term (B) (perhaps "faux-yoichi-question-ism!") for when writers incorrectly mess-up a common figure of speech, and or pathetically accidentally mix one or more together in a meaningless way.
{Aside: For me, these "B" cases are an extremely important and noteworthy social phenomenon in current English-speaking countries: for me it's an absolutely critical pointer to the a hugely significant socio-historic phenomenon playing out currently: a combination of (i) staggering decrease in learned literacy {due to decades of a certain overarching political situation} and amazing decrease in physical native intelligence {due to carbohydrate poisoning}, coupled with (ii) a (rather terrifying) desire to be seen as humorously-pseudo-intellectual. In short, illiteracy + risible peurile pretentiousness == phenomenon "B".}
(A) and (B) are quite distinct, and you in particular very often bring to the fore examples of these. Many of your questions prove to be about (A) (harmless and happy incidents in English) and many are about (B) (which for me are socially disturbing and point to imminent Lord-of-the-Flies -esque societal collapse, on the Jaynesian scale). (!)
In this particular case, it is very much (A). ("Phew!")
A critical point for me is that in the case of yoichi-isms, and more critically, faux-yoichi-isms, answers sometimes confusingly "explain" the phrase without pointing out the overwhelming factor, that it is an (A) {or even (B)} situation.
(So for example, IMO, GMB's answer above is unclear: it relates the situation as if it is a figure-of-speech as such, rather than an "A".)
What does “show some (a little) leg” mean in common understanding?
The information you quote from www.wordreference and www.englishforums. com is correct in describing common usage of show some leg or show a little leg. 
The picture above, from It Happened One Night (1934), illustrates a classic example of showing a little leg as a hitchhiking device. Clark Gable brags about his skill at thumbing a ride; lots of cars go by without slowing down at all. Then Claudette Colbert shows some leg and the driver of a passing car brakes as fast as he can and gives Gable and Colbert a lift.
The idiom or imagery in Dowd's sentence probably is easily understood by a large fraction of the US public, and perhaps by other English speakers as well. However,
Dowd's sentence has problems: It says Holder is trying to appear benign to put pressure on Sterling, which seems to be a self-contradictory statement. That apparent self-contradiction makes the sentence hard to understand, at least during the first dozen times one reads it.
Does [show some (a little) leg] mean to tempt a desired reaction from the other, or to reveal one’s real intention? Is it a common phrase?
It ordinarily is understood as temptation, rather than as a way of revealing one’s real intention, although when Mrs. Robinson in The Graduate shows some leg, it both reveals her intention and acts as a temptation.
Ngrams for show some leg,show a little leg suggests both phrases have greatly increased in popularity since the 1980s, so proper understanding of the phrases probably is widespread.
Best Answer
It's a attempt to shorten the phrase "cooked in some fancy way that a chef might offer in a restaurant, rather than just boiled as a basic dish."
No, it isn't a common term -- I'd call it a neologism, or possibly a term used only by restaurant reviewers. But the "cheffed-up" formulation would probably be understood in this context by most native speakers, by analogy with "prettied up", "tidied up", "polished up" and other common phrases that use "-ed up" to mean "made more so".
Newspaper writers do have a long tradition of writing in a compressed "telegraphic" style, in an attempt to pack as much information as possible into as little space as possible. That does tend to produce phrases which are hard to understand until you have some experience with this style. Don't assume that the New York Times, or any other newspaper, is a good example of formal English.