Do I need a harbor in the capital to make trade routes from other harbors work?

The manual (and the civilopedia) state for the harbor:

Effects: Forms a naval Trade Route with the Capital City, +25% Production of Naval Units, City must be on a coast.

It is unclear to me though whether I need a harbor in the capital to make the trade routes from other harbors work ... ?

The trade network is just that - a network.

In the same way that a city can be linked by road via another city already linked by road - they don't all need their own roads directly to the capital - your trade network will spread via harbours.

What this means is you need at least one harbour within each "sub network" of your trade network. Here is a quick diagram:

Continent A     Water     Continent B
            |~ ~ ~ ~ ~ ~|
     1------2 ~ ~ ~ ~ ~ 3------4-----5
            |~ ~ ~ ~ ~ ~|

City 1 is your capital, and linked by road to city 2. If city 2 and city 3 both have a harbour then they extend the trade route across the water, this means 4 and 5 are also connected, because they are linked to 3 and so, by proxy, use it's harbour link.

If in this example, even if 2 was your capital it would still need a harbour to extend your trade route accross the water, having a harbour in 3 alone is not sufficent.

This also works if 2 and 3 were on the same continent, but not connected by road/rail.

Just to explicitly emphasize a fact which is implied but not expressly stated in DMA57361's and Mad Scientist's answers: the harbour does not need to reside in one of your own cities. An allied city state or a puppet city which has built a harbour will be perfectly sufficient for as long as you have a route to it. They will grant you all their ressources, including trade connections.

With some luck, you can have your continents interconnected without paying maintenance on a single harbour yourself if you connect to friendly (or oppressed) other shore cities.

Not necessarily, you just need one harbor in a city that is already connected to the capital.

There also needs to be a discovered route between the harbors, or the connection won't work. I think also enemy units (or barbarians) can interrupt trade routes, but I'm not sure how that works exactly.