Since this is such a hot button issue one can assume that the language used was proofed many times as not to be used in any other fashion than that was intended.
Given that it is reasonable to make that assumption here. which is there is an observed and significant difference observed as they eluded to the fact they did try to cross reference with other studies though scant as they are.
So no you can indeed feel confident that though not applied in the title correctly the articles conclusion is correct.
If they had evidence that vaccinated people had a higher viral load than unvaccinated people infected with the same variant it would be huge news.
Why wouldn’t they plainly state that?