(source)
Not a map, but at least some more data from some other countries. The own car is unfortunately the most used mode of transport for commuting in every surveyed country, but the US seem to be especially far behind when it comes to alternatives.
I agree that both the map and the statistic I’ve posted don’t take those country-specific characteristics into account.
I’m not sure how important that difference really is, though, as both the US and Germany seem to have pretty similar degrees of urbanization (US: 83.3%; Germany: 77.8%; source). So the rural population isn’t really that big in either country, relatively speaking.
I’m not trying to say that the rural population isn’t a factor, I’m just not sure how big that factor really is.