We believe the Holy Spirit proceeds ineffably from the Father and consubstantially from the Son. There are not "two primary causes" of the Holy Spirit, because all that the Son has is from the Father, who is the sole cause in the Trinity. So the Son's participation in the procession of the Holy Spirit stems from His consubstantiality with the Father. I would hope that explanation makes the meaning of the phrase clearer.
As to Christ and His Disciples - they spoke both Latin and Greek as well as Hebrew and Aramaic, as near as I can tell. I think they would have been able to manage, thank you. For us:
The [Latins] start with the insight into the unity of the Three, that each Person is all of the Divine Nature, so that there is no real distinction in fact between Person and Nature in God, but only between Person and Person in God. But distinction of Person to Person can only be by the opposition of their relations, when you consider the unity of Nature, and that means the Holy Spirit must be from both the Father and the Son or He wouldn't be distinct from either.The Greeks start from the absolute distinction of Persons first and must account for Their unity in terms of origin ... Thus for the Greeks the Holy Spirit must proceed from the Father or the Father wouldn't be the absolute source, and if the Holy Spirit proceeded also from the Son, that would mean the Father ceased being the absolute source. Thus they use two different Greek words for "proceed from the Father": one word for the Son, and another for the Holy Spirit.
Latin has only one verb for "to proceed from," but gets the second idea of procession in Greek of the Holy Spirit (in order to safeguard the Father being absolute source) by adding the words: "as from one principle", and "principally from the Father". Thus there is no contradiction between the Latin and Greek Fathers' teaching ... There is no doubt that the Greeks saw themselves more individualistically and so started with the individual Person with his own absolute personal characteristics as the starting point, while the Latins saw themselves more socially (rationally) and so start from relational concept of person. Thus the Person in the Trinity for the latter is a subsistent relation, while for the East it is an absolute with personal characteristics.
When the Orthodox take the time to listen to the Western understanding, they usually see that it is valid, though it is not their preference.
The Filioque Controversy