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Steps:
-Get any 4 battlecraft leves at some place with the challenge not started.
-Complete a leve
-Get an additional leve (so you don't have to remember the names)
-Complete the 4 leves
-Turn in 5 leves with different nameplates
-Challenge sticks at 4/5 completion.
Are you sure the types were different? Don't think I ever got them all in the same location unless I also put some crafting ones in the mix. Some are restricted to being only offered on a battle, crafting, or gathering class, and it seems like they tend to repeat the same two or three plates over and over for each category. So unless you were changing up your job/class in order to get leves from a different category, it may not have actually been 5 unique leve plates from the same area.
I think you're confused about the requirement of the challenge log. It's not asking you to do Leves with 5 different names, but to do Leves with 5 different face plates. You need to look at the picture that appears on the leve itself in the top right corner of the quest log and make sure that you do Leves with 5 different pictures there.
You can easily get your five different face plates if you have the possibility to do GC leves in your area, since (I believe) all GC leves face plates are different from the normal leves face plates.