The study itself says that the immune response to the vaccine is not as diverse as a natural infection. That's how the study tried to figure out who had been infected:
"Natural exposure induces a dominant Ab response against
the nucleocapsid protein (NP), but since NP is not in the vaccine, there is no vaccine induced response against it. In this way vaccinated people who had a prior natural exposure can be classified because they have Abs to NP."
Simply looking at one aspect of immunity - detectable antibodies in the blood - is a very incomplete way of looking at the overall level of protection. For example, it's well known that mucosal immunity is important, and injected vaccines have a hard time inducing that kind of immunity: https://theprint.in/opinion/why-a-mucosal-covid-vaccine-has-...
It's also concerning that there is evidence that while people who have been vaccinated are much less likely to be infected. If they do get infected, they are more likely to be infected with one of the variants than unvaccinated: https://www.ncbi.nlm.nih.gov/bioproject/PRJNA728463
The immune response developed to our vaccines is quite specific - they only generate the spike protein - which leads to a much less diverse response than a natural infection.
Finally, the study you quoted can't really compare apples to apples anyway, as it's defining prior infection as simply having some level of antibody response, without considering how mild or severe those infections actually were.