Richard is right. The sense in which "culture" is like "cultivate" has no bearing on whether it is a past or present phenomenon. People cultivated in the past, and they cultivate today. They had culture in the past, and culture today. And their current culture, like their current agri-culture, has some continuity with the past, as well as some innovation.

In more tedious academic terms, culture has both diachronic and synchronic dimensions; we can talk about "today's English culture" as a system that exists all at one time, but each element in that system also has a history that stretches into the past. Some elements, like folksong, drama, and literature, stretch back a long way, while other elements, like cinema, only stretch back a few years. Some, like Indian cuisine, have a long history elsewhere and only a short history in England, but they are certainly part of current English culture.

That said, I agree with much of SPB's post as well...I just think the etymology of the word "culture" doesn't go that far in explaining our shared positions.