It says you must make the source available if you distribute the output of the program itself over a network, e.g. in the case of an open source web application

It says you must make the source available if you distribute the output of the program itself over a network, e.g. in the case of an open source web application. Don't you confuse "distribute" and "let users interact with" now? IIRC AGPL doesn't make a distinction about whether i send the bytes of the program over the network (distribute the program) or let a user send input and receive output to/from it (interact with the program).