Don't use demonstratable.
The Corpus of Contemporary American English has 262 hits for demonstrable, and none for demonstratable. Google ngram shows a similar result:
The general difference between "taking" and "taking on" responsibilities in these three cases is:
when you take responsibility for something, it generally means that you accept responsibility and act in an appropriate way by, for example, doing your job (as a parent or a student or a teacher or a banker) as well as you can and don't rely on others to do your work for you.
But
when you take on responsibility for something, it generally means that you assume new responsibility for something you aren't or weren't expected to be responsible for.
There are other ways to use these terms, but all these sentences are about personal and familial responsibilities, not about social or political responsibilities.
"DINK couples refuse to play the role of parents and (take / take on) the responsibilities to cultivate offspring."
DINK couple: "a couple with two incomes and no children". They don't have children, so I'd use take on the responsibilities (because they don't want to have them).
"Mother is getting old, but I haven't (taken/taken on) the responsibility for supporting her."
This is complicated because it's a Chinese cultural problem. If you're the eldest son, then your mother is your responsibility and you must take it (accept it) or violate cultural norms. If you're not the eldest, then you're talking about not assuming (taking on) someone else's responsibilities for your mother. In my wife's family, all six daughters (no sons) contribute equally to the support of their parents, so each has taken on a certain amount of responsibility even though she doesn't necessarily have to, and the eldest daughter cannot afford to support them on her own.
"She is ready to (take/take on) the responsibility of a wife bravely. (Note: Her husband is seriously ill in the hospital.)"
This depends on whether she's a new wife or a veteran wife of this husband. If the marriage is new and the illness is new, then I'd use take on because those kinds of responsibilities are unexpected for new marriages. If, as in my case when my late wife needed someone to be her daily caregiver during the last nine months of her life (she had cancer), after eleven years of marriage, I would use take the responsibility of a dutiful and loving husband. It's part of the marriage vows, but even though almost no one is prepared for such a thing, it seemed to me the only thing to do even though I could have afforded to pay someone else to do it. Had I paid someone else, I would have been shirking my responsibilities, but by accepting them, I wasn't adding anything unexpected to my responsibilities.
Best Answer
Your second definition appears to be a misunderstanding of the standard definition (as amply documented in Sven Yargs' answer). The only "authority" that includes the meaning "take a chance/gamble" for this common idiom is YourDictionary.com, in one of their own (not borrowed) definitions:
This online dictionary is largely an aggregator of definitions, but they also include some "original" definitions written by the LoveToKnow staff (source). I haven't been able to find any information about the qualifications or process of these staff lexicographers. Compare that with this explanation from Merriam-Webster of how words and definitions get into their dictionaries.
In this case, it looks like a writer or writers for YourDictionary have mistaken a phrase meant to convey "come take a look at this [puzzle, so you can help me out with it]" as meaning "come try to solve this puzzle for me."
It's easy to see how such a mistake might be made if the writer(s) in question only encountered the phrase in limited circumstances, and it is possible that some other isolated individuals out in the world have made the same mistake and thus mistakenly use the word this way. It is also possible that this mistaken, but easily accessible, definition will drive usage of this novel meaning of the phrase. However, this is not currently a standard definition of take a gander.
To look for evidence of usage, compare the phrases take a gander at (the usual collocation) with take a gander on (which might be expected at least sometimes for a synonym of take a chance/gamble on):
Now, it is possible that the usage is still at, with a meaning similar to take a guess/stab at. I do find a very few actual examples of take a gander at this/the puzzle/game but all of these still appear to be at least consistent with the sense "have a look" (though, again, someone unfamiliar with the phrase might possibly parse one or two of them differently). For example:
Looking for "a gander at" and gamble returns results like this:
That last example juxtaposes the original usage of phrase take a gander with the very similar take a gamble. Clearly this author knows what the former means, and knows not to use it in the latter's place.
Let me add that these examples are not cherry picked; the first is the only example I could find of "a gander at this puzzle"; I could not find any examples of the phrase with the puzzle, the problem, this problem, or my problem (I don't rule out the possibility that they're out there, but Google wasn't showing them to me). The examples with game and gamble are slightly more numerous (results in the low double digits), but I still didn't spot any examples with the second meaning you cite. I think, from your statement in the question that
you probably haven't had much luck finding examples of the second usage, either.
To make a too-long story slightly longer, I see no evidence that the definition created by YourDictionary.com is an accurate representation of how this phrase is currently used in any major dialect of English. It's possible for it to shift in meaning, but if such a shift has already begun it is so far still essentially undetectable.