Yes, that's the correct meaning to choose, at least I assume so, not having looked at the same dictionary.
I'm not sure their description is entirely right, but yes, as a relative pseudo-direction between places, it can mean going somewhere that's more important, central, prestigious, metropolitan, cosmopolitan, desirable, more of a destination. Something somehow more, better, or more important in some way.
It's complicated by the fact that compass directions also impact choices of word, but when it comes to somewhere like London the "it's the big city so it's up" tends to override it for most of the country. From where I live, I would say "up" to Carlisle, and "down" to Manchester, but probably break about 50/50 between "up" and "down" for London, despite it being far, far south of me (as distances in England go). But anyone in south-east England, or the east of England, and I would guess (with some trepidation) the Midlands as well, will usually refer to London as "up".
Consider the Oxford definitions, which include both "a place perceived as higher" (as in "up to the shops"), but also specifically "towards or in the capital or a major city". Cambridge has, as variants of the same sense, both "towards the north" (with an American example, in fact), and, as a specifically UK usage, "towards a more important place, especially a city".