IBAN, a standard for internationally unique bank account numbers (legally required in the EU but, as I understand it, not adopted in the US) uses a mod-97 algorithm that produces two, not one, check digits.
The Luhn check is definitely used in Visa, Mastercard, and AmEx plus – I believe – Discover. The others out there I have no idea.
It’s a quick and dirty check mainly meant to catch typos in manually entered numbers. It will catch any single digit error plus all transposed digits except 0/1. It might or might not catch a two-digit error.
It also probably acts as a guard to keep people w\from submitting random numbers until the get one that works.
For V/MC, as you alluded, the first six digits identify the issuing bank and what kind of card it is (Wells Fargo – personal, business, debit, etc) , the last digit is the check digit and the nine in between the actual account number.
I was amused on seeing a card, XXXX XX00 0000 123X imagining a small bank with only a couple hundred customers.
Wouldn’t that make that side of the card the front?
I don’t believe so – my Discover card is similar. The only thing on the “front” is the logo, which is why I presume it to be the front. 
The backside, which now has the account number and expiration date, also has the signature box.
Interestingly, one thing I recall from an IT consulting experience with the Visa division of a bank is that the bank cannot just issue any numbers it wants within its allocated prefix. It has to buy blocks of allocations from Visa Inc., and AIUI the blocks don’t come cheap. So all those outrageous credit card fees you and the merchants pay are all partly to pay for those arbitrary charges that are made, essentially, “because they can”.
Not that you should feel sorry for the banks. During the above-mentioned Visa consulting engagement, I had a few dinners at the bank’s “leadership center” (code for “a place to hang out, relax, and pig out on free gourmet food”). We had great meals, but as part of the “lower” echelon, we fetched our own food buffet-style, except for roasts and such that were carved by chefs standing by. But I witnessed a group of portly gentlemen who were described to me as “the Visa delegation” – the top Visa execs from the bank. They got their own private dining room, had their dinners served to them by obsequious waiters, and had a vast wine cart rolled in. It appears that there is money all around in the Visa business! 
I know; I bought 100 shares of Visa company stock shortly after the IPO (prior to which it was owned by the member banks) for about $7000. My theory was that online commerce is sure to grow worldwide and they collect a penny or so of every transaction. I now have 400 shares worth something like $80,000. I only regret not having bought more shares.
Coupla things. Your name isn’t transmitted to the credit card processor, at least not in a card not present situation. I can key a credit card into the my credit card machine and it doesn’t ask for a name. Depending on a few other things, it’ll ask for the zip code, expiration date and, sometimes, the CVV. More often than not, a missing or incorrect CVV won’t be a cause for decline. I assume, at least in part, because so many of them are hard to read. When I’ve spoken to my processor, they’ve generally said that the CVV part really only comes up in the case of a dispute. It’s much more likely to be in favor of the merchant if that number is correct. Also, with some processors, you can tell it to decline if the CVV or zip code is wrong, but it’ll generally work.
FWIW, at least in my line, zip codes are often wrong or unknown because the people with the cards in their hand have corporate cards or third party cards or they’re using their boss’ card and don’t know the billing address.
Now, regarding the expiration date, IME, as long as the month is correct, it tends to still work. I run a lot of credit cards by hand, most of them recurring. I’ve learned that if the card I have on file is expired, ie 07/20, I can enter 07/23 and it’ll generally work. I do, of course, contact the person and update it, but this trick at least means I can finish what I was doing and call them afterwards so I have the correct info for next time.
Now, just to be clear, I always ask the customer for the name/cc number/cvv/exp date/zip code, it’s just that I don’t always need all of it.