None of your scenarios is correct, and some of your premises are invalid too.
First, you don't have an investment of $10K in VTWSX; you own a certain number of shares in VTWSX, and at the current price per share of VTWSX, your investment is worth $10K. The number of shares of VTWSX that you own is most likely not an integer but a number expressed to three decimal places. Now, mutual fund companies determine the price per share of mutual funds at the end of the business day when markets have closed, and all purchases and sales of shares for that day are done at that price per share. (For global investors such as VTWSX for which markets might never close -- somewhere, some place, some stock exchange is open -- the price per share might be determined at a fixed time (say 4 pm in New York). In fact, the price that you see for VTWSX shares on the Vanguard web site is the price as of yesterday. So, what is the price per share that Vanguard determines? Well, VTWSX owns stock shares in various companies, bonds, etc and it may have some cash on hand in its own "checking account". It adds up the value of all the stock shares and bonds etc that it owns, using the price of the most recent transaction on the exchanges for each item, plus the cash in the checking account, and divides this by the total number of VTWSX shares owned by the investors in VTWSX to determine the price per share for that day.
If you have sent in your request to get $1K from your VTWSX investment before 4 pm New York time today, Vanguard will execute your order by redeeming exactly enough shares of VTWSX (fractions to three decimal places, remember?) at the price that will be determined as of 4 pm that day (though Vanguard doesn't post the price per share on its website till about 9 pm) to generate $1K in cash. It will send you the money the next day by wire transfer (expensive) or by ACH transfer to your linked bank account (free but it might take two or three days to get the money) or by paper check (free but it will take them a day or two to print the check and then there is the transit time via USPS). What will be left in your account will be whatever shares of VTWSX that were not redeemed, and the value of your investment will drop not exactly by $1K; the value might be more than $9K or less than $9K depending on whether today's price per share is more than or less than yesterday's price per share.
So, what's the role of this "checking account" that VTWSX allegedly has? Well, that's the account into which all the money accompanying purchase orders is deposited, and it is the account from which redemptions are paid out. VTWSX will rarely sell shares of stock that it owns to pay redemptions (almost certainly not for a $1K redemption) unless there is a "run" with lots of people redeeming VTWSX shares and very few buying; all the immediate transactions are done in the "checking account" and actual buying and selling of investments that VTWSX owns occurs only in bull markets (lot more money flowing into VTWSX) and bear markets (lot more money flowing out of VTWSX).
Note: all the above applies to VTWSX, the traditional mutual fund, and not to its ETF counterpart VT.