Here is an example of what the CDC counts as vaccinated and unvaccinated.
Effectiveness of COVID-19 Vaccines in Preventing Hospitalization Among Adults Aged ≥65 Years — COVID-NET, 13 States, February–April 2021
Link takes you to a table
https://www.cdc.gov/mmwr/volumes/70/wr/mm7032e3.htm?s_cid=mm7032e3_w#T1_down
The footnotes of the table define the terms:
“ Partially vaccinated patients received 1 dose of Moderna or Pfizer-BioNTech vaccine ≥14 days before hospitalization or 2 doses, with the second dose received <14 days before hospitalization.
Fully vaccinated patients received both doses of Moderna or Pfizer-BioNTech vaccine, with second dose received ≥14 days before hospitalization, or receipt of a single dose of Janssen (Johnson & Johnson) vaccine ≥14 days before hospitalization.”
Studies are looking for vaccine effectiveness, not vaccine issues. If there are issues, that would require getting the spreadsheets and looking at when people got sick after taking vaccines, and seeing if there is a pattern with factors like age, comorbidities, time after vaccination of infection.
From figure 2 in the link you posted, partial vaccination has issues. If people are getting sick after the first does or shortly after the second dose in a narrow date range, then would be a cause for concern. AFAIK no one has presented the data on when people are getting sick during and after vaccination. Any doctor raising these issues risks censorship like Dr. Charles Hoffe.