Only luggage for travel.
A backpack would be "una mochila".
A shopping bag would be "una bolsa de la compra", usually just "una bolsa" if the context is clear.
Additionally, a briefcase can be called "un maletín" or "una cartera"; without context, cartera is ambiguous, because it can also refer to a wallet.
I'm not sure what you mean by "goes the other way". I'm a New Zealander, so my English is more British than American, and I've lived in the UK.
I don't think I've ever heard anyone use either baggage or luggage as a count noun. I would never refer to, say, "ten baggages/luggages" - I'd always use the singular to refer to a collection of any number of individual items.
Is that not standard usage in both British and American English?
Right, I think I've figured out the source of confusion.
Collective nouns (like band or government) are indeed treated differently in British and American Englishes, and there's a good explanation on Wikipedia:
But luggage is (I think) a mass noun, and those are always grammatically singular.
When you say "every piece of baggage...", the noun is "piece", which is definitely countable and singular, so I don't think it would ever be acceptable to use "are" in that case.
The question is whether you could ever say "the baggage are...", and I'm not entirely sure of the answer, or whether there's a US/UK difference. To me, "the baggage is..." seems more natural in every case I can think of.
Your example with the band is a bit different, in that it's a group of people, and they could each individually begin to play, so the singular and the plural can both work. It's a bit hard to think of a case where each piece of baggage acts individually.
I used to watch Craig Ferguson on American television and now I watch Doctor Who on BBC America. On those shows they say things like "CBS are," "the government are," "UNIT are," etc. It used to really grate on my nerves because in America we would use "is" as the verb in each of those instances. Now that I have seen so much and understand they are using proper UK English, I've gotten over it.
An American would say "The band begins to play," but I have read British literature that says "The band begin to play."
I know that an American would say "Every piece of baggage is . . . " I suspect that it was once grammatically acceptable for a British speaker to say "Every piece of baggage are . . ."
The difference is in whether baggage and the verb that agrees with it are considered singular. It is possible that my understanding is outdated or entirely mistaken.
I am quite confident that "the band begin to play" is not acceptable in American English.
Perhaps the example I gave could have been better - but I was trying to get across the idea that a singular noun used to describe a group of things (like "band," and maybe baggage) is always considered singular in American English, but occasionally plural in British English.