Translation:No worries! Goodbye!
"It doesn't matter" would be the most accurate translation in my opinion. 没关系 is a bit more formal than the more commonly spoken 没事 which directly translated would mean something like no matter. If you look up mei guan xi in a chinese dictionary the translation options are.... it doesn't matter, it's nothing, that's all right, never mind. So "that's ok" should be accepted....however "no problem" is translated as mei wen ti 没问题. In my opinion, "no worries" would be more related to that. The reality is this all semantics but the answer should encompass ALL correct answers!! Especially for such a common and basic phrase!
All of these phrases are conceptually synonymous and thus perfectly acceptable translations of each other. Being pernickety about only accepting exact, literal translations of common phrases that express the same meaning is more likely to confuse new learners, in my opinion.
yes, I think 'no worries' has been migrating to the US from Australia. I was born and raised in Southern Illinois and never heard 'no worries' until my siblings who migrated to Australia came to visit with their children born in Australia. Now I spontaneously say 'no worries' and it seems more common in the US.
对不起 méi guānxi - This has a few translations in the dictionary: it doesn't matter; it's alright, never mind, don't worry.
Like 'no worries', 没关系 can be used to either dismiss a worry or to say you're welcome (sometimes the concept could be one in the same). The best answer to your confusion is to learn more vocabulary to be more specific: 没问题 (méi wèntí) literally means 'no problem', 不用谢 (Bùyòng xiè) meaning 'no thanks necessary' is an excellent "you're welcome". Nevertheless, I've found that one of the beauties of Chinese is the artful way of communicating effectively while being purposefully ambiguous.
The word 关系 (guanxi) means "relationship", but in the way of "relationship points" you get when you buy someone a gift, cook them dinner, or do someone a favor (the contrary is also true, when you call favors you "use" your 关系). That means once someone does a favor to you it creates an unspoken obligation to repay that favor. Once you and someone else build up 关系 you have a relationship.
Considering that 没 is a word of negation, 没关系 actually means "you don't owe me any favors", so, there is no reason to thank.
Yes it is confusing to translate Chinese characters. You can not always make a literal translation. Take it as a vocabulary lesson, because you will learn an english word that did not translate from the characters.
Yes, 'no worries' is typically Australian. However, it is being used more often in the U.S.