Not "more likely to experience another COVID-19 infection". That is what the Israeli study looked at - taking all positive test results in the country (as in people who tested positive - Israel has a health system that can tell if one person was tested more than once) then sorting out which were from people who had been a. fully vaccinated at the time with no known previous covid infection. b. completely unvaccinated with previous, separate covid infection. c. everyone else. Then compared the first two catergories.
The CDC study looks at all the people admitted to seven hospitals who had covid-like symptoms. Then takes odds adjusted rates of a. fully vaccinated at the time who had no known previous covid infection. b. completely unvaccinated people with previous, separate covid infection. c. everyone else. Then compares each the first two categories - what percent of each tested positive to covid.
It may be a subtle difference. Or not, I'm not sure at this point. Either way, it follows the pattern of trumpeting conclusions that aren't what the given evidence supports.
I haven't figured out why they conclude 5.49 fold higher odds when their results are 9% of unvaxxed and 5% of vaxxed. - of people hospitalized for covid-like symptoms - not percents of people who have natural immunity or of people who are fully vaxxed.