Oh geez, Bill Warren. His work has made its rounds on /r/AskHistorians in the past-- /u/shlin28 and co. did a good job addressing his methodological and conceptual errors, as well as his 'battle database'. in a previous thread. /u/Valkine and others have also addressed similar claims. You may find their answers helpful as well.

I have also written on medieval justifications of crusade. I should note that while medieval theologians and crusaders certainly reasoned armed pilgrimage as an act of defense on behalf of Christians and Christian sites, framed within a cosmic struggle between good and evil, it does not follow that their narrative must accurately reflect reality.

Warren somewhere along the line got it into his head that medieval Islam was a singular entity, a monolith that acted with one purpose. Any act of violence therefore must be an expression of jihad because Islam is a hive-mind, and the actions of small groups must be in accordance with an overarching goal or master plan. Or something absurd like that. Between 632 and 1095, however, there is no evidence of a persisting plan among the various polities that spanned the Islamic world to consume Europe. Warren wants to make the debate about Islam vs. Christendom, but what we know about medieval European and Middle Eastern society does not reflect that narrative.

I am not going to suffer through that whole video, nor is it worth my time to draw up a seratim refutation, but, shockingly, the man with a math PhD is a terrible historian.

To the point you raise: were the Crusades a reaction to Jihad?

The short answer is no. The long answer is nooooooooooooooooooooooooooooooooooooooo. Although some versions of the speech delivered by Pope Urban II at Clermont, which is usually credited with launching the first Crusade, spend some time on various outrages committed by Muslims. Some mention the pressure being put on the Byzantine Empire. But, for a variety of reasons, none are uncontroversial reproductions of the speech itself. Moreover, there are no real precipitating events on part of Muslims. It's true that al-Hakim, the Caliph of Egypt, destroyed the Holy Sepulcher, but in 1009. Yes, the Byzantines had been crushed in the Battle of Manzikert... in 1071. The Clermont speech was given in 1095.

I would recommend Jonathan Riley-Smith's The Crusades as a good intro text, if you'd like to read deeper.