I would not sweat the name confusion thing. This is done all the time, and neither your current 401K company nor Vanguard has any interest in pilfering that money. Do it the way everyone else does it and it will go smoothly. If, by some slim miracle of circumstance, it doesn't go smoothly, there are mechanisms in place to track the money down and though it might be a small PITA, the right outcome will happen.
I would also, to the maximum extent possible, use Vanguard as the ones to initiate the process. (Use the firm that is "pulling the money" from the other firm as a general guiding principle. They have more experience and interest in making it smooth than the firm that is having the money taking from their management, plus Vanguard does 100s of these transactions per day with a variety of counterparties, so they're good at it. Each "pushing bank" does only a few per day with each specific counterparty. The path of least resistance is to start with Vanguard and follow their instructions. Don't think too hard about them, just follow them to the letter.) This is probably a large and scary sum to you to have in flight. That's normal. These transactions are done many, many times per day, in amounts far larger and circumstances more complex than yours, and the custodians are skilled and effective at making it happen.
Your funds will be sold and the assets transferred as cash (a check conveys cash, not securities) and then invested in Vanguard funds you choose. This is what you want anyway. (There are other means to transfer accounts and holdings without selling them. There is usually no point to doing this in a tax-advantaged account anyway. You want to move the money as money and have Vanguard re-invest it.)
I would roll the 401K money into a distinct IRA that only holds 401K rollover money. I would roll the Roth 401K into a distinct Roth IRA account that only holds Roth 401K rollover money. This protects your flexibility to later roll those funds into a future employer's 401K/Roth 401K plan. Once you comingle those rollovers with regular IRA contributions, you either can't roll them into a future 401K or it is an inordinate PITA to do so. (I think it's not possible, but I'm not 100% on that.) Though I have no plans to do such a rollover (because Vanguard IRAs are generally better fee structures than employer 401Ks), I also found it didn't cost me anything to keep them separate. If your assets at Vanguard are low enough that they're charging you $20/account/year fees, then you might want to consolidate them into an existing IRA to minimizes the number of accounts on which you're paying those fees.