You are correctly parsing the information.
There are two categories of duration.
Running on its own power (solar) and other supplies, it can handle a week of free flight as your source indicates. Not sure what the specific limiting factors are. Possibly the solar arrays are not large enough for a complete charge, but are enough in combination with the batteries to keep cycling for a week.
When docked to the ISS it can get power from the station at which point the limiting factor is other things. How long before the arrays degrade due to being in space?
The first Crew Dragon demo flight has only a 110+ duration we were told during the launch videos and meetings. This is apparently due to a simpler solar array that degrades quicker in space.
The production Crew Dragon vehicles will have better arrays that can handle 200+ days to meet the NASA requirement.
The Soyuz vehicles have about a 270 day duration docked, which is due to fuel issues. (I forget if it is peroxide or hydrazine and seals).