Does this mean buying locally is in some sense equivalent to buying from another economy?
It depends on how bilateraly opened are the economies of the countries that are in trade interactions. To get an idea of which country ultimately benefits the most from those trade interactions, the sign of the Uk-to-SKorea-trade-balance is likely to be a good proxy. A positive sign would mean that wealth redistribution due to these trade interactions are greater locally than abroad. But still, remembering that wealth concretly consists of incomes stemming from labor and dividends, its redistribution is strongly related to having (or not) world-wide shareholders and local workers (that are also consumers) that mainly (in value) buy goods locally. And if what they buy locally is actually also mainly produced locally, i.e. the Uk-to-World-trade-balance is positive, we can reasonably think that wealth redistribution mainly occurs at the domestic level.
Why is the consideration of trade deficits/surpluses relevant?
Trade deficits/surpluses are good proxies of how netportable (somehow exportable $-$ importable) the aggregated purchasing powers of coutries in trade interactions are. Indeed, the idea is that the difference between buying locally versus abroad vanishes if owning/accessing foreign markets is as easy as owning/accessing local markets -- except for workers who would be shareholders of nothing, which is rather unlikely: we all have a savings account(?) Not that obvious.
In the scenario in my question the ten pound note ends up sitting in the account of someone backing the foreign exchange broker.
The ten pount note is an asset that has its own value in South Korea's currency. If you assume that currency exchange markets are perfectly efficient, there is not point in considering transitory states and only the final usage matters. Thus, the definition you make of final (and transitory) is of first importance and is contextual to a specific question, with an explicitly formalized time horizon. Moreover, when sitting in the account of someone backing the broker, the ten pound note can be used to create money if, say, it is short-sold or lended.
Can we agree that the ten pound note will be either at rest in a repository (bank account/under a bed) […]
Incidentally, even the fact of having it really at rest in a repository is very unlikely to be neutral : this will have impact on the solvency of the depositor. With a higher solvency she can benefit from lower interest rates and/or from better leverage effects.
Can we agree that the ten pound note will […] be spent in the UK economy (as that is the only accepting economy)?
The point is that this ten pound note may stay "unused" for long periods of time and simply be considered as a traditional asset, sold here and there. The amounts that would be collected during sales could be used all around the world. However, is this an observed behavior ? I.e Is British pount used as a safe haven value? If the answer is No, then indeed a UK note is very likely to be spent locally as that is the only accepting economy.
At best I see a difference in the speed of money with respect to spending locally vs abroad. […] So the only real difference between local and abroad is efficiency of spending (fees, speed of money).
IMO, this summarizes well the idea $+$ how bilateraly opened are the economies of the countries that are in trade interactions. On the other hand, and strictly speaking about the 10 pound note as such, this additionally complexifies our story by outlining that if this speed is "low" and relative-to-dollar inflation "high", the note of 10£@year0 may worth ~0.00£ in year0-based real terms many years later.