There's like a shit ton of sources on that wiki, though.

The pope at the time wanted something for the future to remember him by and when Byzantium asked for some help, the pope used it to raise a crusade and take over the holy land for Christianity. It was about prestige and power.

I don't really see how any of this is relevant to the discussion we were having. In what you said, you concede that Muslims took over those lands (from Romans). Why wouldn't Byzantium, by virtue of being part of the Roman empire, want those lands back? Who cares if it was 300 years later? Also, the Muslims conquered the lands with years of terror and bloodshed. Didn't they sort of kind of deserve to be kicked out? Isn't this the progression of history? Tribe conquers tribe conquers tribe?

At the point of the Muslims capturing Jerusalem in 637, they had essentially crushed the Byzantines into pulp. In like 40 years they beat back the Byzantines in (using contemporary names here) Beirut, Palestine, Israel, Egypt, Armenia, Jordan and most of Syria. The Byzantines didn't have the manpower for another war.

Either way, what I think should be conceded at this point is that history is murky as shit. I don't think either of us can know the exact intentions of the Romans or the Muslims.

I do think this is an important point stated from the wiki, though:

"Generally, subsequent historians have either followed Erdmann, with further expansions upon his thesis, or rejected it. Some historians, such as Speros Vryonis, have emphasized the influence of the rise of Islam generally, and the impact of the recent Seljuq onslaught specifically. Steven Runciman argued that the crusade was motivated by a combination of theological justification for holy war and a "general restlessness and taste for adventure", especially among the Normans and the "younger sons" of the French nobility who had no other opportunities"

Basically, historians disagree about this shit. People who have studied this their entire lives disagree about this shit. You and I both having taken "a few" or "two" Crusader history courses means jack shit and isn't a credible avenue of argument.