"those who have received three doses of a vaccine and test positive for COVID-19 are more likely to be infected with infections compatible with the Omicron variant compared with those who are unvaccinated"
All this is saying is that P(Omicron variant | positive test AND 3x vaccine) > P(Omicron variant | positive test AND no vaccine).
In no way this is implying P(Getting Omicron | 3x vaccine) > P(Getting Omicron | no vaccine).
"Vaccination status: those who have received three doses of a vaccine and test positive for COVID-19 are more likely to be infected with infections compatible with the Omicron variant compared with those who are unvaccinated, though individuals who had received at least one dose of a COVID-19 vaccine continued to be less likely to test positive for COVID-19, regardless of variant. It is too early to draw conclusions from our data on the effectiveness of vaccines against the Omicron variant."
I note that it says "It is too early to draw conclusions from our data on the effectiveness of vaccines against the Omicron variant." also.
Does it mean more likely to get infected (given the same exposure) or more likely to be infected? The difference is quite important.