Sadly, I am poorly equipped to answer your primary question. I sincerely hope that one of our more qualified historians can answer you, but I'm uncertain as to whether we have evidence to say. However, I would like to briefly address one of your secondary points.

especially considering a lot of them were likely just peasant levies who could barely afford the equipment.

Everything we know about 12th century warfare, and the Third Crusade in particular, argues against this statement. The general levy of freemen had, in fact, become very uncommon on the continent by about 1000 CE. Warfare increasingly became the domain of the semi-professionals: whether landed aristocrats, household men, organized mercenaries, volunteers, or organized urban militias. Even in England, where the levy persisted until the 14th century, it was a limited levy of the well-equipped classes; the various regulations imposed by Henry II and his successors make clear that the very poor were not expected to participate in warfare. It was also more commonly used for defensive warfare or local campaigns of limited duration: say, in order to confront the most recent Scottish invasion of Northumberland. We don't know how exactly Richard I recruited his infantry, but given his continental upbringing and the geographic scope and duration of the campaign, I am strongly inclined to think he relied on paid volunteers rather than compelled militiamen.

I go into the nitty-gritty of the military side of the Third Crusade in more depth here. But to comment briefly on Richard's infantry, our evidence indicates that they consisted of two types: heavy infantry equipped with large shields, spears, and felt and/or mail armor, and archers and crossbowmen who supported them. Half-equipped peasant levies don't really fit either bill; heavy infantry equipment ain't cheap, and likewise the crossbow was generally a specialist's weapon. Our sources also indicate that the Crusader infantry impressed their Muslim adversaries with their discipline, both on the march and in battle, and their stoic willingness to endure heat, thirst, and enemy harassment, which again point to more professional troops.

Physically wearing a cross was certainly a sign that set apart a crusader from a normal pilgrim. The vocabulary that developed for speaking about crusaders was the crucesignatus, one who is marked by the sign of the cross. In the 13th century, one preacher encouraging his audience to take the cross, said:

Such people are given the sign of the cross as a sign that they take up this war for the faith of the Crucified, and as a sign that they are soldiers of the Crucified carrying his sign, and as a sign that the large indulgence which is granted them is taken entirely from the treasure of Christ’s passion, which was filled up on the cross.

When Bernard of Clairvaux preached in favor of the 2nd Crusade in 1146 at Vezelay, supposedly there was such enthusiasm for it that they ran out of pre-cut cloth crosses, and Bernard gave up his own cloak to continue - indicating that, regardless of the veracity of that particular detail, that it was common and expected that crusaders would physically wear a cross sewn onto their garment.

I can really only comment on this subject tangentially, as my reading in the subject concentrated on theological themes in how the crusades were preached, but in that preaching, "taking up the cross" was probably the most common and persistent image used. One account of the Council of Clermont in 1095 where Pope Urban II declared the 1st Crusade includes this detail:

He instituted a sign well suited to so honorable a profession by making the figure of the Cross, the stigma of the Lord’s Passion, the emblem of the soldiery, or rather, of what was to be the soldiery of God. This, made of any kind of cloth, he ordered to be sewn upon the shirts, cloaks, and byrra of those who were about to go.

So at first, at least, it doesn't sound like it was necessarily red cloth used for the crosses, as this records "any kind of cloth" and it was affixed to whatever clothing was appropriate. Of course, in later crusades, the image of "taking the cross" became a synecdoche for the entire undertaking of taking an oath, leaving on crusade, etc. and the crusade preachers certainly milked this for all it was worth - but at the root of that, it still seems that there was a literal wearing of the cross on one's clothing. To circle back around to the main specific question, the image of a set uniform of white tunic and red cross seems to come from the Knights Templar, who did use that, at least among some knights and in part of their history, as a distinctive emblem.

Sources:

• Christopher Tyerman, God's War: A New History of the Crusades

• Edward Peters, The First Crusade: The Chronicle of Fulcher of Chartres and Other Source Materials

• Christoph T. Maier, Preaching the Crusades: Mendicant Friars on the Cross in the Thirteenth Century

If you were on the walls of Jerusalem in 1099 looking out at the army of the First Crusade, you would very likely not have seen a signal individual tabard in white with a red cross.

The earliest crusades had no established ritual for joining the crusade, so it was a very ad-hoc procedure that only sometimes involved taking up and marking yourself with symbols. For example, two texts, Robert the Monk and the anonymously author of the Gesta Francorum, detail how in 1095 Bohemond, seized by religious fervor, marked those who joined him with crosses torn from fabric, probably spare clothing, that he had on hand. Robert adds a further detail lacking in the Gesta: such was the desire that Bohemond actually ran out! But, I would hesitate to say that forces that reached Jerusalem still had such markings. Maybe some of the knights whom Bohemond granted these fragile, incongruous tokens maintained them over the army's long journey, but it is not a fact remarked upon by any of the chroniclers and, even if so, his company was just one of many disparate groups that made up the First Crusade.

For most of the twelfth-century, importantly there is no word for crusaders; chroniclers refer to them as Franks, Christians, and other banal terms. By the end of the twelfth-century, however, the word crucesignatus (trans: marked by the cross) began to emerge as a synonym for someone who had taken a crusading oath. There is no evidence that I am aware of that people took this term literally, it is at its core a legal definition: being a crusader came with certain privileges. Indeed, it is far more common that knights would have taken pride to display their own heraldry. Jumping to the middle of the thirteenth century, Joinville, a crusader and courtier in the company of the French King Louis IX on the the seventh crusade, constantly comments on the decoration of shields and other heraldic displays as a means of identifying fellow knights, both in battle and at court. When the Count of Jaffa arrives to greet Louis IX's forces in Egypt, Joinville remembers in an extended passage the beauty of the vessel, whose sides were painted with the heraldic shields of the Ibelin house. Coincidently, their arms also take the form of the cross.

By the time of the Third Crusade, amongst the forces of the crusaders you might see knights or sergeants in the dress you describe. But, these would have been members of established military orders. I am not certain if members of military orders were ever given crusading privileges—calling historians of military orders!—, but they would be in that dress whether they were participating in the activities of the crusade or not. Such dress signaled their vows taken in the service of their order.

I don't know much about the composition of Richard's army, so I will defer to /u/Rittermeister on that point. But, I will add that Conor Kostick has recently described the First Crusade as "a slice of medieval society on the move": full of lots of non-militarized people upon whom medieval communities depended.

For the term crucesignatus, the classic article is Markowski, "Crucesignatus: Its Origins and Early Usage" (1984).

edit: spelling is hard.

A crusader might not actually be in white with a red cross but generally he would have some kind of cross, usually a cross-shaped cloth sewn on his clothing, as a sign of the vow that he had “taken the cross.” (Medieval people did not have a verb “crusade” or a descriptive noun “crusader.” The French “croisade”—from the Latin “crux” or “cross”—seems to first occur in the late 15th century; the OED dates the earliest English use of these terms to the early 18th century.) William Chester Jordan explains these physical marks of the vow in his 2009 lecture “Crusader Prologues: Preparing for War in the Gothic Age” (with his citations included if you wish to follow them):

Crusaders were men who had ‘taken the cross’ or ‘been signed by the cross’, which is to say, they had publicly received a cloth cross that they wore as proof of their vow to go and fight. . . .

I have already mentioned that one who vowed—publicly declared—to go to war under the aegis of the cross received either a cruciform cloth or else strips of cloth to be sewn in cross-form on an over-garment. This reception occurred simultaneously with the public avowal or soon afterwards.^1 Rarely, a crusader opted to be branded or incised with a small emblem of the cross on the shoulder, much as soldiers now get tattoos that announce their comradeship with their fellow men and women and their commitment to their cause.^2

1 William Jordan, Louis IX and the Challenge of the Crusade: A Study in Rulership (Princeton: Princeton University Press, 1979), pp. 13, 216; Christoph Maier, “Civilis ac pia Regis Francorum deceptio: Louis IX as Crusade Preacher,” in Dei gesta per Francos: Etudes sur les croisades dédiés à Jean Richard, ed. Michel Balard and others (Aldershot, UK: Ashgate, 2001), pp. 57-63.

2 Riley-Smith, “State of Mind of Crusaders to the East,” p. 68; John Shinners, comp., Medieval Popular Religion, 1000-1500: A Reader (Petersborough, OT, Canada, and Orchard Park, NY: Broadview Press, 1997), p. 165.

EDIT: Added the subtitle of Jordan's lecture.