The President negotiates. The Senate approves. So Kerry is correct that the Senate cannot 'modify' the agreement, other than by getting the president to add terms the Senate will find acceptable. But it is nothing more than a suggestion or opinion until approved by the Senate. So the 47 are correct that the next president will not be bound by anything unless it is approved by the Senate. IMHO
Article 2 Section 2 "He shall have Power, by and with the Advice and Consent of the Senate, to make Treaties, provided two thirds of the Senators present concur...
“The President negotiates. The Senate approves.”
Actually the “Advice” part provides that the Senate provides input into the treaty and not merely acts for ratification.