I understand that the Quranic concept of jihad refers to a quite different spiritual struggle - what I'm interested in is the politics and warfare of Islamic realms during the High Middle Ages. From what I know of the history of the Crusades, responses to them were relatively fragmented, weren't they, with local states often allying with the Christians against traditional enemies?
This is the kind of topic that really requires an entire book to do it justice - and someone with more specialist knowledge about Islamic law and theology than I have - but I'm going to do my best to give at least the beginnings of an answer to it. Before I get to the specifics of your question, I want to talk a little bit about the tendency in popular culture to view Jihad as the Muslim equivalent of the Crusade, and the huge can of worms that idea opens.
The core problem with even engaging with this idea is that – as is so often true in crusades studies – the very ideas we’re debating don’t reflect contemporary conceptions of what was happening. The very word “crusade” is an early modern invention, nobody in the 11th, 12th, or 13th centuries called what they were doing “crusades”. This means that any attempt to create a unifying concept of what exactly constituted “A Crusade” is a modern construct we project back onto the past. Generally, historians view “legitimate crusades” as those declared by the papacy during the period between 1095 and 1300 with the purpose of conquering or supporting the holy city of Jerusalem and its environs. How far beyond that definition you go depends on your take on what constitutes a “crusade”. Does campaigning in Spain count? What about the Baltic? What about killing heretics in Southern France? What about early modern campaigns against the Ottoman Turks? There’s a lot of room for debate here, and historians can be narrow or broad in their definition of “crusade” and that’s fine. The problem here is that when you have a flexible ahistorical definition like that, how can you then go on to define something as it’s opposite or “X’s equivalent” of it?
Jihad doesn’t have quite the same complications, in that it is at least a contemporary term that was used in the Middle Ages. The problem, as you mentioned in your question, is that Jihad in the Qur’an generally refers to a spiritual struggle, and it doesn’t lose that meaning in the Middle Ages, it just gains more possible meanings. This creates a lot of complications where historians have to interpret whether a religious writer means Jihad in an internal sense, or maybe just a metaphorical sense (we must all “struggle against the infidel” kind of idea), or in a literal take up arms and drive them out sense.
Then you have to consider that, initial papal schism aside, most of what we would consider the Main Crusades (the ones with numbers) were declared by a single, central, recognised spiritual figure (the Pope), while Islam was much more fragmented at this point.^(1) So in this inverse world, who is responsible for declaring a legitimate Jihad? A major part of what makes a crusade into A Crusade is the Pope, but there isn’t really a Muslim equivalent to the Pope (for complicated reasons I can’t go into here, the Caliph doesn’t really fulfil the same role, although even then for much of the 11th and 12th centuries there were two competing Caliphs in the Holy Land, so even then it’s complicated).
This is all to say that this stuff is complicated and weird, and to even make a comparison between these two concepts we kind of have to make a lot of assumptions, some of which are probably misleading or inaccurate. These are essentially two very different things that just kind of overlap a little bit sometimes.
The really short version of the answer to your question is that no, these games don’t accurately represent either movement. The Crusades and the “counter Jihad” (if we want to call it that, I’m not sure we do) were inherently fragmentary movements. Neither one represents the entire religious community rising up as a unified front to battle against the other. Both groups had insane levels of internal fragmentation. Saladin – probably the most famous Jihadi general of all – spent more time fighting other Muslims than he ever did Christians. Similarly, it’s impossible to study the Crusades without discussing the petty internal politics of the campaigns, whether it’s the eternal squabbling of the princes on the First Crusade or Philip II’s decision to leave the Third Crusade early in part so he could get back to Europe and continue undermining the power of the English monarchy in France (despite ostensibly being Richard I’s ally against Islam). Religious divisions did exist, and it’s impossible to discuss this period without discussing religious violence, but they didn’t entirely eclipse political rivalries and personal ambition.
When Muslim powers did effectively fight back against Western invaders they primarily did so in the form of major political leaders and dynasties. Historians speak of the Muslim response in terms of its leaders – people like Zengi, Saladin, and Baibars – and their families rather than any proclamations of religious war declared by leading spiritual figures of the time. While it wouldn’t be accurate to completely divorce the military careers of these leaders from their spiritual beliefs, it’s also not accurate to see them as subservient to some religious fanaticism that drove them against the Western Christian elites. It’s all more complicated and messier than that, like basically all of history ever really.
^(1) I’m totally leaving out the Byzantine/Western church schism here because Crusading is essentially a Western European phenomena – although the Byzantines obviously weren’t entirely divorced from it.
Hopefully that at least kinda answered your question, it was all a bit more rambly than I hoped but it’s actually a really tricky question to answer right, and as I said to do it its full justice would take a few hundred pages. With that in mind, I’ve recently finally started reading Brian Catlos’ Infidel Kings and Unholy Warriors and while I’m not finished with it yet what I’ve read so far has been a really interesting examination of the intermixing of political and religious affiliation during the Crusading Period. Probably worth checking out if you’re interested in learning more.
In more general history, I’m always a big fan of Thomas Asbridge’s The Crusades, and if you want a Muslim perspective Paul Cobb’s The Race for Paradise is excellent, although it’s more political than theological in its focus.
My answer is more related to the Western Crusades, the Iberian Reconquista, but I think it can elucidate some points.
In the Iberian peninsula, the call for Crusades (and there were many) didn't usually get the support of foreign non-Iberian kings, as the Crusades for the Holy Land or Egypt did in the East, but lots of counts and lower nobles from France, England, Flanders, etc participated in it.
In the same way, the calls for Jihad that I know of were actually answered by people living in the same political entity, or, at most, their neighbors.
When the first taifas (independent small realms that came after the first Iberian Ummayad Caliphate collapsed) called for the North Africans Almoravids for help in defending against Christian incursions in the 11th century, they framed it as a Jihad. So that can count as a Jihad answered by a foreign ruler. Of course, it differed 5 years later, because the Almohads outright conquered the taifas.
But, as you pointed out yourself, the situation in Iberia was much more complex than a muslim x christian conflict, with everyone aligned in their camps. There were many, many conflicts during Iberian history that had muslims or christians in one side against muslims or christians in the other side, with one side or the two sides mixed up.
In the battle of Las Navas de Tolosa, for example, the Almohad army had some Christian mercenaries on its side. It also had a large contingent of Jihad volunteers. Most of them came from the Almohad territory. Their imam-caliph had called for the Jihad, ordering the imams and other religious figures to do so in the temples in the territory they controlled. So a good number of people inside that territory answered. From the sources of the time, these Jihad volunteers were irregular infantry and put in the front in the order of battle, basically to "soften" the enemies. The core of Almohad army was made up of professional soldiers and tribal levies, as well as the mercenaries mentioned earlier.
It must be pointed out that the Almohads (and the Almoravids, to a lesser degree) were practicing a form of islam that was kind of different from the one that was practiced in Iberia before. If you want to simplify things, they were more "fundamentalist" / "intolerant" than what was there before, with the Almohads especially using the rhetoric of reforming Islam and being much more intolerant towards non-Muslims. The Almohad caliph, as well, kind of centered Jihad on himself, i.e. only he could do it. This actually made some peace treaties with Christian kingdoms actually last longer, since before, when Jihad was "decentralized", the truces were broken a lot by some random lord on either side.
Sources:
Pascal Buresi. The story of the Almohads in the Kingdom of Fez and of Morocco
GUERRA E PODER NA EUROPA MEDIEVAL DAS CRUZADAS À GUERRA DOS 100 ANOS, Monteiro, Martins, Agostinho (2015), Universidade de Coimbra
IBN IDARI. Al Bayan al-Mugrib. (original source published in 1325. I'm working with the 1965 Spanish translation)