Western Catholics also recognize that nos. 6 and 7 apply to all bishops. Our claims for the bishop of Rome are that he exercise this kind of authority granted to all bishops but also stands at their head for the sake of a focus permitting unity to be maintained. His role of "confirming the brethren" (his brother bishops) is simply that--not lording over them but at their head for the sake unity leading the universal episcopal college and serving as the ultimate point of reference for resolving differences among them because unity is so important (Jn 17).
That, then, is the sense in which no. 8 applies. Adhering to Peter and his successors is another way of saying that the college of bishops, all of whom are authorized by Christ to govern in their dioceses in ways included under 6 and 7 and in a lot of other ways, voluntarily submits, in case of genuine disagreement among them, to Peter's successor for the sake of unity.
Cyprian already made clear that each bishop is a Peter in his own diocese and that they colletively bear responsibility for governing the Church, but that in the face of real schism and unresolved disagreement, Peter's successor as bishop of Rome has a special role, for the sake of unity. Exactly how that role gets played out (the details of Petrine jurisdiction), that's what we are arguing about most of the time. And it is not carried out the same way vis-a-vis the Roman rite as it is vis-a-vis the Eastern rite Catholics. And we recognize that it can benefit from further clarification. We just won't let go of the basic claim.
"We just won't let go of the basic claim."
At this point in history, or so I have been informed, Orthodoxy isn't asking you to, Dion.