Based on my non-linguist opinion, I would say you're correct. Dutch people tend to not capitalise their Ds in surnames. When only mentioning the surname, it is treated as a mini-sentence, hence the capitalisaiton.
In northern Belgium (AKA Flanders), Dutch speakers tend to capitalise their Ds in surnames, so you would see "De Zwart" regardless of where it appears in the sentence. In case you're wondering: yes, there are quite a few surnames that exist both in The Netherlands with "de" and in Belgium with "De". The same goes for "van" (of) and "vander/van der" (of the), for instance.
Then again, if they're nobility, they might not capitalise the "de" because it might be the French "of" as opposed to the Dutch "de", meaning "the". Confusing stuff.
It makes sense: when you say Juan del Río it's obvious that it's a surname, but if you just said "del Río" without context, the "del" could be confused with the common word.