What Do Trinitarians Mean By A Divine "Person"? (Quotes)

Here are some statements from Trinitarians that
illustrate the tension obtaining between modalism,
Trinitarianism and tritheism.

"For, in truth, as the Father is not the Son, and the
Son is not the Father, and that Holy Spirit who is
also called the gift of God is neither the Father nor
the Son, certainly they are three. And so it is said
plurally, 'I and my Father are one.' For He has not
said, 'is one,' as the Sabellians say; but, 'are one.'
Yet, when the question is asked, What three? human
language labors altogether under great poverty of
speech. The answer, however, is given, three
'persons,' not that it might be [completely] spoken,
but that it might not be left [wholly] unspoken"
(Augustine of Hippo. On the Trinity 5.9).

"It is generally admitted that the word 'person' is
but an imperfect expresson of the idea. In common
parlance it [the word "person"] denotes a separate
rational and moral individual, possessed of
self-consciousness, and conscious of his identity amid
all changes. Experience teaches that where you have a
person, you also have a distinct individual essence.
Every person is a distinct and separate individual, in
whom human nature is individualized. But in God there
are no three individuals alongside of, and separate
from, one another, but only personal self-distinctions
within the Divine essence, which is not only
generically, but also numerically, one" (Louis
Berkhof. Systematic Theology. London: Banner of Truth,
1971. Page 87).

"But to say nothing more of words, let us now attend
to the thing signified. By person, then, I mean a
subsistence in the Divine essence, - a subsistence
which, while related to the other two, is
distinguished from them by incommunicable properties.
By subsistence we wish something else to be understood
than essence. For if the Word were God simply and had
not some property peculiar to himself, John could not
have said correctly that he had always been with
God.When he adds immediately after, that the Word was
God, he calls us back to the one essence. But because
he could not be with God without dwelling in the
Father, hence arises that subsistence, which, though
connected with the essence by an indissoluble tie,
being incapable of separation, yet has a special mark
by which it is distinguished from it" (John Calvin.
Institutes of the Christian Religion 1.13.6).

"If these [i.e. the three persons] are taken as three
separate centers of consciousness in an
individualistic way, as some modern thought seems to
do, then one would end up with tritheism, a denial of
the Trinity. Equally, one can overemphasize the unity
to the detriment of the persons" (John Thompson.
Modern Trinitarian Perspectives. New York: Oxford
University Press, 1994. Page 6).

"We have employed the term 'person' and must
unavoidably do so, but hopefully we have indicated
that this word should be used of Father, Son, and
Spirit with full awareness of the analogous nature of
this predication, therefore with stress on the radical
distinctiveness of 'person,' not just differentiating
divine from human personhood, but distinguishing the
reality of personhood as it applies to Father, Son,
and Spirit" (Bernard Cooke. Beyond Trinity. Milwaukee:
Marquette University Press, 1969. Page 58).

See also http://fosterheologicalreflections.blogspot.com/search?q=o%27donnell