I am the author of the web page in question.

The claim you're making is that Manwë put some kind of limitation on the eagles: some types of help are acceptable, but other types are prohibited. As far as I am aware, the text does not say this anywhere. The argument you make is indirect: it is based on the types of help which we observe the eagles actually giving.

If Manwë did put some sort of general limitation on the forms of help which eagles are permitted to give, how exactly could that limitation be formulated? We know that the eagles can carry the ringbearer (they carry Bilbo at a time when he is in possession of the Ring). We also know that they will fly into Mordor. The limitation by Manwë would somehow have to permit those things individually but rule out doing these two things at the same time.

Since there is no direct evidence in the text that Manwë enacted any such limitation, I think that the simpler account is that no such limitation exists.