You're contradicting yourself. If the Sievert already has units of a rate (J/s), then the 400 mSv per hour you mention is a double rate (energy/time^2), some kind of energetic acceleration, which doesn't make sense here. Your second paragraph is correct, but it contradicts your first.
As others have noted, the units for a Sievert are J/kg, not J/s. This is a very important distinction. An accumulated does requires these units, as J/s is a rate, and then you have to know how long a person is exposed, i.e. there is no accumulation. An accumulated dose implies that if you receive 1 mSv, that is all one needs to know: there is no time scale involved. It is a certain amount of total radiation received. Correcting your first paragraph, 1 Sv received in 1 second is (approximately) the same as 1 Sv received in 1000 seconds and as 1 Sv received in one million seconds. Sieverts are therefore a useful measure for directly determining the effects the radiation will have on a person.
So in fact Randall's image is accurate, unless there is some minor error in it that hasn't yet been discovered. Given your own misunderstanding of the situation, I hope the press's confusion is a little less inexplicable. You still come to the correct conclusion, which they often do not, but sometimes, science is hard.