Who determines this - the individual college’s Admissions Officers? My specific interest is whether a Jewish applicant would potentially be considered a URM?
No
Every college (not the individual AO) where not prohibited by law can use it’s own definition of URM. In general, URM is African-American, Native American/Native Alaskan/Native Hawaiian, Hispanic/Latinx. Some colleges might expand out to include Native Samoans or other Pacific Islanders. Some might include First Nations of Canada.
To the extent that a university/college wants to give special consideration (added bump) to a particular minority group, that group would be a URM for the purposes you are talking about. This would be an institutional and not an individual AO call.
There could be schools that want to increase the number of Jewish matriculates like schools that want to increase Asian Americans, who are otherwise ORM’s for other schools, but I think that would be rare.
For colleges that use URM considerations, the group considered is usually African Americans, Hispanic Americans, and Native Americans. Many also consider women as URM’s for engineering. Those applying as internationals usually do not get URM treatment. Some colleges may apply it to others as noted above and some officially do not apply it (but use other methods such as considering economic differences). I am not aware of any college that considers Jews as URM.