I would like to add some theological background to this answer from the Catholic perspective (and naturally, I would invite Orthodox readers to contribute their own perspective).
First, some historical background:
As the original question points out, the Nicene-Constantinopolitan Creed was originally written in Greek. Although the fundamental concepts in that Creed were worked out by the year 381 in the First Council of Constantinople, the text that is commonly used today in liturgical celebrations, both Eastern and Western, does not seem to have stabilized until the Council of Chalcedon, in the year 451.
(The Council of Nicaea, in 325, as regards Trinitarian dogma, was only concerned about the relationship between the Father and the Son, since the divinity of the Son was contested by Arius; it was the Council of Constantinople I that first explicitly affirmed the divinity of the Holy Spirit.)
In any event, as was pointed out, the text in Greek that was eventually universally adopted contains the phrase
[Πιστεύομεν] εἰς τὸ Πνεῦμα τὸ Ἅγιον ... τὸ ἐκ τοῦ πατρὸς ἐκπορευόμενον.
[We believe] in the Holy Spirit ... who comes forth from the Father.
This is the text as found in the acts of the Council of Chalcedon. Note that the phrase τὸ ἐκ τοῦ πατρὸς ἐκπορευόμενον echoes John 15:26, which says in Greek
τὸ πνεῦμα τῆς ἀληθείας ὃ παρὰ τοῦ πατρὸς ἐκπορεύεται ἐκεῖνος μαρτυρήσει περὶ ἐμοῦ. (The Spirit of Truth, who comes forth from the Father, he will bear witness to me.)
The various Vetus Latina and Vulgata versions of the New Testament translate the word ἐκπορεύεται as procedit. This translation is probably OK for the ordinary language of the time, but it causes a problem in a technical theological document such as a symbol of faith. If you observe the etymologies of both the terms, they have subtly different meanings: the verb ἐκπορεύομαι means "to come from (ἐκ) something;" the Latin procedo means "to issue forth (pro)."
The key difference, at least the way the different traditions understood the terms, is that ἐκπορεύομαι insists on the ultimate origin of whatever has come forth, whereas procedo only insists on the fact of issuing forth.
To give an analogy, suppose the President of the United States sends a letter to the Prime Minister of the United Kingdom. The President, naturally, sends it through his Secretary of State. In Greek, if you ask "from whom does the letter come forth (ἐκπορεύεται)?" the answer is "the President, and the President only." In Latin, if you ask "from whom does the letter 'proceed' (procedit)," you could truthfully answer, "from the President and from his Secretary of State," because the important thing is that it has been issued forth.
In more technical terms, whenever the Eastern Fathers spoke of ἐκπόρευσις, they always understood it in such a way as to refer to the monarchy of the Father. Only the Father can be the ultimate Source or Origin of the ἐκπόρευσις. No Greek Father ever said, "the Spirit comes forth (ἐκπορεύεται) from the Son;" they would rightly consider such a statement to be heresy, since it would imply that the Son is also an ultimate Source, and hence does not receive everything He has from the Father.
The Western Fathers, on the other hand, developed their Trinitarian theology using the language first developed by Tertulian. The Western Fathers understood "procession" to mean the communication of the Divine Essence from the Father to the Son, and from the Father through the Son to the Holy Spirit. For Latin Trinitarian theology, the concept of "procession" does not automatically imply that the principle of that procession must be the ultimate source.
Both Eastern and Western Father's agree that the Father communicates His essence to the Son, and that the Son communicates that essence to the Holy Spirit; the Western Fathers call that communication "procession;" the Eastern Fathers do not call it ἐκπόρευσις, but rather use a different term, τὸ προϊέναι.
Why did the Western Church insist on the insertion of the Filioque? In the Latin version of the Creed, the equivalent clause says
[Credo] in Spiritum Sanctum ... qui ex Patre (Filioque) procedit.
As I mentioned, for the Latin Fathers, procedere in this context means "to communicate the Divine Essence," the same as the Greek τὸ προϊέναι. If one were to deny the Filioque, it would be tantamount to denying that the Son communicates His Essence to the Holy Spirit, which is contrary to the universal teaching of the Fathers.
Moreover, it would imply a type of subordinationism: if the Son is truly "constubstantial" (one in Substance or Essence) with the Father, then He has received everything that Father has (except Fatherhood, which is a relation of origin). That includes the capacity, so to speak, to communicate that Essence to the Holy Spirit. Denying the Filioque would imply that the Son has received the Divine Essence only "partially" (if that makes any sense) and hence that He is not fully consubstantial with the Father.
We can say, in summary, that the Western theological tradition has, for historical and linguistic reasons, fused together into one concept called "procession" what Greek theology has divided into two: ἐκπόρευσις and τὸ προϊέναι. Neither approach is erroneous, but each one must be understood in its context.
Therefore in Greek, the following statement is heresy:
τὸ Πνεῦμα τὸ Ἅγιον ἐκπορεύεται ἐκ τοῦ Πατρὸς και τοῦ Υἱοῦ (the Holy Spirit comes forth ultimately from the Father and the Son).
Likewise, the following statement in Latin is heresy:
Spiritus Sanctus procedit ex Patre tantum (the Holy Spirit proceeds only from the Father).
(Interestingly, the Eastern Catholic Churches—the churches in communion with the Bishop of Rome that employ Eastern liturgical practices—do not use the Filioque when they recite the Creed, and the Catholic Church has insisted that they do not.)
My conclusion, after researching this question extensively, is that there is no fundamental contradiction between the Eastern and Western understanding of the Trinity, merely a difference in approach and terminology.
(The basis for this reflection can be found in an excellent document entitled the Greek and Latin Traditions Regarding the Procession of the Holy Spirit, issued in 1995 by the Pontifical Council for Promoting Christian Unity.)