Does the J1 visa really require 51% of the funding to come from the hosting institution or an external source?
J1 is a broad category of visa, covering from camp counselors to Post-Doc students. Each category has a different goal and requirements. It looks like that your question is related to being a "college/university student".
According to the 22 CFR 62.23 (the legislation):
A student is eligible for participation in the Exchange Visitor Program if at any time during his or her educational program in the United States:
(1) The student or his or her program is financed directly or indirectly by:
(i) The United States Government;
(ii) The government of the student's home country; or
(iii) An international organization of which the United States is a member by treaty or statute;
(2) The program is carried out pursuant to an agreement between the United States Government and a foreign government;
(3) The program is carried out pursuant to written agreement between:
(i) American and foreign academic institutions;
(ii) An American academic institution and a foreign government; or
(iii) A state or local government in the United States and a foreign government;
(4) The student is supported substantially by funding from any source other than personal or family funds; or
(5) The student is participating in a student internship program as described in paragraph (i) of this section.
That is why the links that you sent require that at least 51% of the funds come from an institution.