You can use 'quite' in more than one context. Both of those contexts given make sense and are used in British English. I don't think there is a distinction between US and British English, although it would be nice if an American or a Canadian could confirm that.

You are "quite" right. Americans use it the same way.

Yes, that is 'quite' correct. There is a difference between the way that British and American English speakers use the word 'quite'. But it isn't 'quite' as simple as the explanation you give. The meaning number 2 in your definition is used in both the American AND British English. In British English, the word 'quite' can serve one of two functions, depending on whether it is modifying a gradeable or non-gradeable adjective. Gradeable adjectives are words like 'tired' or 'big' or 'small'. Non-gradeable adjectives are words like 'exhausted' or 'enormous' or 'tiny'. 1. He is quite tired = He is fairly tired/ He is a little bit tired This is an example of 'quite' used with a gradeable adjective. It softens the adjective, and means 'a little but not very'. 2. He is quite exhausted = He is totally exhausted/ He is absolutely exhausted. This is an example of 'quite' used with a gradeable adjective. It strengthens and emphasises the adjective. British English has both meanings of 'quite' : numbers 1 and 2. American English only uses meaning number 2.

It's the same in both.

Both meanings [1and 2] are used in UK English, depending on the context. 'quite a lot' means 'a lot' - 'quite' is added for emphasis 'quite a few' or 'a fair number' of something or some people means a substantial proportion of them : 'Quite a few people in the place speak English' It is not as many as 'most', but more than 'a few' I can't comment on American usage. UK use of this word is, as you can see, rather difficult to understand. Just keep your ears open for examples.