Umm, leves DO differ based on the graphic.
For example, Valor leves (Showing a lancer attacking a dragon) simply require defeating a set number of foes.
Tenacity leves (Showing an elezen on a chocobo) will have the last enemy run off for reinforcements.
Diligence (Showing a lalafell holding two dead marmots) requires getting a set number of items.
Wisdom leves (Showing a bard, with an ogre in the background) are all the Necrologos leves, requiring you to get a set number of pages either by defeating foes or checking glowing spots, which summon a boss after each set.
Veracity leves (Showing a lalafell leading a dark blue... roegadyn I think?) involve finding the true target that's hiding as an innocent. Fighting the target forces a change in all versions, though some allow you to collect items to reveal targets without engaging them.
Temperance leves (Showing a mi'quote carrying a jug of water) involve heading to various areas on your map and defeating enemies that spawn as part of a patrol. This leve currently seems bugged, as you do not need to defeat the enemies for a patrol to finish, meaning that the boss that spawns after finishing all the patrols will spawn while you're still fighting the last group of monsters. Only the boss need actually be defeated, despite the leve descriptions making the patrol mobs the actual targets.
Promptitude (Showing... someone mounted on an antelope? Purple and white color scheme) are exclusively for company leves, and involve defeating as many opponents as you can during the course of the leve.
Piety (Showing a Mi'quote at a dock with a fishing pole) is exclusively for fieldcraft leves, and involves depleting a given number of nodes of their resources.
Munificence (Showing a lalafell carrying a large basket of goods) is exclusively for fieldcraft leves, and involves getting a set amount of a given leve-specific item.
Justice leves (showing a balding man kneeling in front of a bearded wise man) are battlecraft leves, but I've yet to do one of these so I can't specify what their focus is.
Ingenuity leves (Showing a woman handing out bread) are local leves. You receive your materials immediately, and your target is a given aetheryte camp.
Industry leves (Showing a man panning for gold) seem to all be Horn and Hand faction leves. They involve no combat, merely going around and examining glowing spots to collect materials.
Hability leves (Showing a lalafell leading sheep) also seems to just be for Horn and Hand faction leves. It involves following someone or something while avoiding detection, so you can claim what it leads you to.
Devotion leves (Showing a red toned gladiator) involve going to a specific area and killing all monsters that approach. Leve is completed when you defend the area for long enough, even if monsters are still alive.
Constancy leves (Showing a roegadyn working a forge) are the other local leve type, and differ from ingenuity in that you need to talk to an NPC to get materials first. The benefit of these leves is they are all done in the city (With the exception of the level 1 leves that involve helping out the innkeeper, these are done at the level 1 camp), which can save on travel time.