Badges were more general. National badges like the cross of St. George might be worn by all who were members of, or who served, that nation. Household badges might be worn by all those that were part of, served, or supported a great household.
Devices were usually more specific. They might be an emblem chosen by a single individual for limited purpose: such as a single man who sought a single opponent for a deed of arms. Often they were only slightly broader: a small group of companions who sought opponents willing to fight on equal terms.
Liveries allowed great and middling households to identify their members and allies. The most closely affiliated supporters got clothing of particular colors and a badge. Less closely affiliated supporters got the badge alone. Livery clothing was often of more than one color to increase the number of clearly distinct liveries.
Some urban militias also wore uniform clothing. Froissart describes the Flemish army before Roosebecke:
The men from each town or castlewick had similar uniforms (parures semblables) so as to recognize one another: one company wore a coat made of blue and yellow, another a black band on a red coat, another white chevrons on a blue coat, another wavy stripes of green and blue, another checkered black and white, another quarterly white and red, another blue with one red quarter and another cut with red above and blue below.The London watch of 1378 distinguished groups of wards by the color of their lances: white powdered with red stars, all red, white environed or wreathed with red, black with white stars, and all white.
Jupons were also worn over armor in the 14th and 15th century. The term seems to have generally referred to garments without heraldic arms and so distinguished from coat armor, although they might be covered with brocade or other patterns.
When did men-at-arms wear coat armor, and when did they wear something else? It's a complicated question. On the one hand, men-at-arms wanted their individual valor and prowess to be recognized. On the other hand, coat armor was expensive, or fragile, or both.
Also, not all men-at-arms had arms to display. In 1389, John Kingston was thought worthy enough to sustain a challenge from a French knight, but he then had neither arms nor formal rank as a squire, and so both were granted by letters patent.
Further, it seems that "divers men" on the Agincourt campaign assumed arms for the campaign that they had neither inherited nor been granted. Henry V's band of brothers were allowed to keep their assumed arms, but in 1417 he prohibited assuming arms unless the bearer "possess or ought to possess the same in right of an ancestor or by gift of one having sufficient power."
And not all of the men entitled to bear coat armor on the battlefield always did so. 14th and early 15th c. manuscripts frequently show men-at-arms bearing arms on their shield or horse trappings but none on their body, and frequently a jupon of an entirely different color than any in their arms. The mid 15th c. Beauchamp Pageant shows the Earl of Warwick in coat armor for most of his battles, but in white harness fighting against the forces of Owen Glendower in 1402, identified only by his crest of a bear and ragged staff.
14th and 15th c. manuscript illuminations almost always show only a minority of the men-at-arms in a battle wearing coat armor.
While marching infantry are shown in livery coats, I've seen no clear evidence in iconography of men-at-arms wearing them in battle. Two apparent exceptions appear in MS M.804 a version of Froissart's chronicles from ca. 1412-1415. It turns out that both fol. 338r and 347v are showing Flemish urban militia as described above, but drawn as men-at-arms with full leg harness because that is the artist's default way of portraying soldiers.
One should be reluctant to draw conclusions from iconography alone. Fortunately, the closely related Agincourt accounts of Jean Le Fèvre and Jean de Waurin shed light on who wore coat armor and when.
To tell the truth, the king of England had wanted to lodge in another village which had been taken by his herbergers, but he, who always observed proper and honourable practices, did what you will now hear. It is true that whenever he wanted to send scouts before towns or castles or any matter, he had the lords or gentlemen take off their coats of arms when they went off and put back on again when they returned. It it so happened that on the day that the king left Bonnières to go up close to Blangy, there was a village which had been commandeered by his harbingers, but he had not been told of it. Not knowing in which village he was supposed to lodge, he went on by a bow shot and rode past it. Then he was told he had passed it. Then he stopped and said 'As I have passed, god forbid that I should return as I have got my coat of arms on'. And he moved on and lodged where his vanguard was lodging, and moved the vanguard further forward.From these accounts it appears that coat armor was particularly associated with pitched battles, and that anything that could be considered retreat could be considered dishonorable once it was put on. For that reason Henry did not want it worn when scouting, because the men would necessarily have to return to the main body, and this could be described as retreat. Also, coat armor was worn not only by commanders, but by at least some of the ordinary gentlemen.
...But to return to the king of England, before he crossed the river at Blangy en Ternoise, because the crossing was narrow he had six bold men of his vanguard take off their coats of arms and cross over in order to find out whether the passage was guarded. They found that there was no one seeing to its defence, so they crossed quickly.
Later, Le Fèvre and de Waurin tell how Anthony, duke of Brabant, rode in such haste to the battlefield that he left the main body of his men behind.
As he would not wait for them, because of the haste with which he had come he took one of the banners from his trumpeters, made a hole in the middle of it, and used it as his coat armour.So, the duke did not have coat armor with him on the march, but thought it so important to wear it in battle that he made improvised coat armor from a banner.
Participants in deeds of arms for single or group combat by mutual consent probably wore coat armor even more frequently than on the battlefield. The phrase "his coat of arms on his back" recurs with monotonous regularity in accounts of these combats. The exceptions tended to be individuals who wore sumptuous finery designed for that particular deed of arms, as when Jacques de Laing wore "a robe of sanguine silk all strewn with blue tears" for one of his combats at the pas of the Fontaine des Pleurs, or to make a political statement, as when Juan de Merlo wore "a vermillion-coloured mantle, with a white cross on it, like to the badge of the French" in 1435.
Jousters might wear their personal coat armor, but often wore team uniforms, often designed for a particular joust, or other garments according to their fancy.
Tournaments, in the traditional sense of mounted melee combats, had become quite rare by 1350 everywhere outside Germany.
King René, when he attempted to revive the tournament west of the Rhine ca. 1460, expected that coat armor would be the norm for noble participants. Illuminations of tournaments ca. 1350 are more ambiguous, with some participants wearing their arms on their clothing and others wearing clothing completely unlike their arms.
Curry, Anne. 2000. The battle of Agincourt: sources and interpretations. Woodbridge: Boydell Press.