There's no misinformation.
Loyola is an independent private school that is NOT run by the archdiocese. Admission is granted through ISEE/SSAT standardized testing, which is the same test used to determine admission to the other independent private schools that have no religious association, like Dalton, Horace Mann, and Riverdale Country Day. The ISEE/SSAT is also used to determine admissions to other PRIVATE schools with non-Catholic religious instruction. Trinity (EPOSCIPAL) requires it. Friends Seminary (Quaker) also requires it.
Admissions to the high schools that are part of the archdiocese is done through the COOP examination and the COOP is only used for Archdiocese Catholic School admission.
Religious orders has nothing to do with anything. St. Vincent Ferrer (CATHOLIC ARCHDIOCESE) is under the Dominican Order. Xavier High School (CATHOLIC ARCHDIOCESE) is under the Jesuit Order. Loyola High School, while Jesuit like Xavier, is NOT an archdiocese school run by the Catholic Church. It is an independent PRIVATE school.