Marc said it best here:
Couldn't have said it better myself (at least I was trying to earlier).
Like I said before, these same compliments that are given to Rome were given to others, like Constantinople and Alexandria.
And to answer your other question. I don't think the East knew or thought Rome was in charge. If anything, they may have brushed the few attempts of Rome's supposed supremacy as nothing but bullying. But the East and Rome may have misunderstood one another. But it's quite telling that Pope Gregory the Great apparently was humble enough of a Pope to acknowledge that all other bishops have the same universal honor as he does, as he mentioned to the Alexandrian Pope from the quote I gave to you earlier.
But in any case, please share with us any quote that we can discuss.
This is the last quote I found here, then I'll rest so you will have time to answer:
St. Maximus the Confessor (c. 650)
A celebrated theologian and a native of Constantinople, ...
The extremities of the earth, and everyone in every part of it who purely and rightly confess the Lord, look directly towards the Most Holy Roman Church and her confession and faith, as to a sun of unfailing light awaiting from her the brilliant radiance of the sacred dogmas of our Fathers, according to that which the inspired and holy Councils have stainlessly and piously decreed. For, from the descent of the Incarnate Word amongst us, all the churches in every part of the world have held the greatest Church alone to be their base and foundation
, seeing that, according to the promise of Christ Our Savior, the gates of hell will never prevail against her
, that she has the keys of the orthodox confession and right faith in Him, that she opens the true and exclusive religion to such men as approach with piety, and she shuts up and locks every heretical mouth which speaks against the Most High. (Maximus, Opuscula theologica et polemica, Migne, Patr. Graec. vol. 90)
How much more in the case of the clergy and Church of the Romans
, which from old until now presides over all the churches which are under the sun? Having surely received this canonically, as well as from councils and the apostles, as from the princes of the latter (Peter & Paul), and being numbered in their company she is subject to no writings or issues in synodical documents, on account of the eminence of her pontificate
even as in all these things all are equally subject to her according to sacerodotal law
. And so when, without fear, but with all holy and becoming confidence, those ministers are of the truly firm and immovable rock, that is of the most great and Apostolic Church of Rome. (Maximus, in J.B. Mansi, ed. Amplissima Collectio Conciliorum, vol. 10)
If the Roman See recognizes Pyrrhus to be not only a reprobate but a heretic, it is certainly plain that everyone who anathematizes those who have rejected Pyrrhus also anathematizes the See of Rome, that is, he anathematizes the Catholic Church. I need hardly add that he excommunicates himself also, if indeed he is in communion with the Roman See and the Catholic Church of God ...Let him hasten before all things to satisfy the Roman See, for if it is satisfied, all will agree in calling him pious and orthodox. For he only speaks in vain who thinks he ought to pursuade or entrap persons like myself, and does not satisfy and implore the blessed Pope of the most holy Catholic Church of the Romans, that is, the Apostolic See, which is from the incarnate of the Son of God Himself, and also all the holy synods, according to the holy canons and definitions has received universal and surpreme dominion, authority, and power of binding and loosing over all the holy churches of God throughout the whole world.
(Maximus, Letter to Peter, in Mansi x, 692).
If you have similiar quotes attributed to any of the eastern sees during the time period of the early church as the above, I'd love to see them.