Learn English – Are constructions like “That’s me out, then” primarily British rather than American

Way out and exit mean approximately the same thing. Is it too simple an answer just to postulate that our authorities may have happened to standardize on different terminology because it sounded better to them, or because it sounded more naturally-spoken in the respective country? In other words: there isn't a 'reason'. You're overanalyzing.

American English is a living language spoken and written by millions of people in the year 2014, that continues to evolve.

British English is a living language spoken and written by millions of people in the year 2014, that continues to evolve.

Both have features and vocabulary that the other once had and since lost (or at least which is now less commonly found in it).

Both have innovations that the other has not adopted as eagerly.

Describing either as archaic or modern compared to the other is meaningless (now Yola for example, is a form of English that is genuinely not modern).

It could certainly make sense to describe one as more innovative in a particular regard, but in actually examining the two we find that the two seem to keep apace for the most part.

There's perhaps more spellings that differ from how they were in 1801, due to Webster's reforms being more heavily adopted in the US than the UK, however:

1. Many of these were a matter of him settling on one of two or more forms found in both the US and the UK, so in these cases neither is necessarily the more modern.
2. Many were adopted in the UK too.
3. Many were not adopted in the US.
4. There were spelling innovations in Britain such as beginning to favour -isation over -ization.

We find verbs changing forms more strongly in one than the other, but it will sometimes be British English that is the innovator, sometimes American.

We find many neologisms in American English, but also some relics like teamster being used long after any teamster dealt with horses.

A great many differences relate to concepts or inventions that are themselves relatively recent, and hence the term for either is equally recent in both.

A lot of terms have come into one of these countries from its immigrant populations and its imperial adventures, but different terms have come into the other from its different immigrant populations and different imperial adventures; while US soldiers may have become "gung ho" later than than for UK soldiers missed "blighty", there isn't really much justification calling one more of an innovation than the other.

Really, while one could spend time producing a thorough score card and argue one way or the other on the basis of it, in any meaningful sense they're both about equally modern.