Learn English – What does “apostrophe s” stand for in “What’s it all mean”

It's not about a contraction "winning" over a possessive. "Its" is the possessive form of "it", like "his" is of "he", "her" is of "she" or "their" is of "they". There is no missing apostrophe; the forms go back to a time when English was a highly inflected language. It predates modern, or even Middle, English.

The possessive formed by the apostrophe+s construct is a more modern, uninflected, less-marked form. There are only a very few commonly used words — pronouns — that still use the older forms. Markedness tends to survive in words that are used very frequently, even when other aspects of the language are losing their markedness. It's the same reason why we still say "men, women and children" rather than "mans, womans and childs" when the plural ess marker is nearly universal in the rest of the language.

No, it is not a short form of anything. Here, will is not an auxiliary verb, but a full verb. Nothing is omitted in the sentence. Will, here, is used in the meaning "want" or "wish", which is considered archaic in most other contexts, outside of set phrases. It is related to the German wollen, Dutch willen, etc., all with the same meaning "to want, desire".

will

 1. (obsolete) To wish, desire. [9th-19th c.]

if you will

 2. So to speak

Here is a related Language Log post that makes a rather interesting point of "if you will" being semantically strikingly similar to the ubiquitous Valley-Girl like.

Like functions in younger speakers' English as something perfectly ordinary: a way to signal hedging about vocabulary choice — a momentary uncertainty about whether the adjacent expression is exactly the right form of words or not. If the English language didn't implode when if you will took on this kind of role among the baby boomers, it will survive having like take on an extremely similar role for their kids.

Emphasis mine.

Edit: here's yet another Language Log post that actually discusses a comic strip that unfairly criticises "if you will" precisely because the cartoonist mistakes will for an auxilliary verb.

Language Log goes on to comment,

In his latest intrusion, it's refreshing to see Lemont [Brown, the protagonist] identify himself as a member of the Idiom Police, since his too-literal reading of "if you will" has nothing to do with grammar. [...] What "if you will" does not do, of course, is require the interlocutor to "accept the words you just chose to say," as Lemont claims. I suspect Lemont's the type of non-Gricean guy who responds to the discursive filler "you know" with "No, I don't know."

It then elaborates on that, linking to the aforementioned like post and quoting the definition from UsingEnglish as well as the one from OED:

Oxford English Dictionary says ["if you will"] is "sometimes used parenthetically to qualify a word or phrase: = ‘if you wish it to be so called’, ‘if you choose or prefer to call it so.’" Citations are given for the exact phrase back to the 16th century, with similar elliptical uses dating back to Old English.