Ok, I'm going to try to explain this one more time, and please do read what I wrote in the previous post again, slowly this time.
There are several types of logical criteria that characterize propositions. First, they are either well-formed or ill-formed, they are either coherent, or incoherent, they are either consistent or inconsistent. If we were to suppose that a proposition were the conclusion to a logical argument, then we characterize the proposition (and thus the argument,also) as logically valid and/or logically sound.
So, a proposition is "illogical" if it is ill-formed, or incoherent, or inconsistent, or logically invalid, or logically unsound.
This is how
logicians use the word. You are using the version of the word they use in Florida panhandle trailer parks. Naturally, because there are more people speaking like Florida panhandle trailer park occupants than speaking like logicians, the dictionary records the more popular usage. Hey, even this year the dictionary added "
twerk," "
srsly," and "
fauxhawk."
So, you go ahead and use the word however you like, and we'll know exactly what to think of you by what you choose.