I'm fairly certain there's a record of the Basileus specifically asking for western support from Urban himself.
All the accounts for what was actually said in the Council of Clermont greatly differ on "why" the council actually happened. Also, their is a huge debate between scholars about how important the Council of Clermont actually was. Personally I agree with the more recent research which argues that the First Crusade was a reaction to apocalyptic concerns many Christians had with the dawning of a new millennium, in which case it is reasonable to suggest that an armed pilgrimage eastward would have happened even if the Council of Clermont hadn't been called.
The Chronical of Lupus Protospatarius wrote: (This is my translation from the Original Latin, I will post both)
1095- De mense Aprilis in nocte quinta feria subito visi sunt igniculi cadere de caelo quasi stellae per totam Apuliam, qui repleverunt universam superficiem terrae. Et ex tunc coeperunt Galliae populi pergere, immo totius Italiae, ad sepulchrum Domini com armis, ferentes in humero dextro cruces vexillam.
During the month of April on the night of the fifth day (Thursday) fireballs were suddenly seen all around Apulia falling from the heavens like stars, which filled the surface of the earth. And then the peoples began to proceed out of Gaul, indeed all of Italy, to the Sepulcher of the Lord with arms, carrying in their right hands a banner of the Cross.
Notice, there is absolutely no mention of the Council of Clermont or helping the Byzantines. The author of this Chronicle says the beginning of the first crusade was because of a violent meteor shower. I have found the same mentioning of that meteor shower in chronicles from all over Italy, France, England, and even Prague. Many dont even mention the Council of Clermont, other mention the Council of Clermont only after talking about the meteor showers.
Here is a Chronicle from England by Orderici Vitalis: (Once again, this is my translation from the original Latin, but I dont have that text right now)
In the year of our Lord 1095 a day before Nones of April it was seen by an innumerable amount of spectators from Gaul, many stars running about. It was believed to be like a hail-storm, but of dense light. Many of these stars were falling, as explained in the Scripture, which says that, when stars fall from the heavens (Matthew 24:29).” The particular passage reads, “"Immediately after the anguish of those days, the sun will be darkened, the moon will give no light, the stars will fall from the sky, and the powers in the heavens will be shaken.”
This idea of the First Crusade being motivated by fear of the Apocalypse is advocated by Dr. Jay Rubenstein, who teaches at the University of Tennessee, and wrote the book "Armies of Heaven: The First Crusade and the Quest for Apocalypse."
So, yes, I would argue that helping the Byzantines doesn't have much evidence to support it and that even the importance of the Council of Clermont has been greatly overstated.
Also, if you can read Latin you can look at a ton of Medieval primary source material at:
http://www.documentacatholicaomnia.eu/