I did answer your first question directly. Hopefully we can get this right the second time around...
Yes, the meme is specifically directed towards people who ignore or completely dismiss other viewpoints just based on the assumption that any information available that is counter to their own beliefs or understanding is wrong.
Some people have a fear of the potential for being wrong or have too much pride which makes them biased.
If a person comes to a final conclusion by obtaining "X" amount of information, then that's fine but if they want to confirm with 100% certainty that their beliefs and conclusions are correct then they will have to be open to learning everything about "Y" information as well.
If this person "doesn't know" about all the information available they aren't close minded, they're just ignorant (aka lack info). To state that a person "doesn't care to know" suggests a few things about that person. For instance, they might be lazy, too busy to learn, close minded, lack interest in the subject matter or suffer from confirmation bias.
You wrote..." What if the person disagreeing with you does, in fact, have all the information necessary to come to the conclusion they've reached (whether you agree with that assessment or not), and simply disagrees with you...?"
This meme isn't about "competing" over who is right versus who is wrong. It's about being as objective as possible and the only way to do so is to obtain all the information at one's disposal before making any final judgement. Still, obtaining all the information available doesn't guarantee a person will be correct.
Regardless of whether it's about objective truths or matters of opinion, the meme still stands. If they choose to not be open to alternative information then they are limiting their own knowledge.
You Wrote..."What if BOTH parties have all the information necessary to come to their respective conclusions, and still disagree...?"
Then most likely BOTH sides have limited information and/or refuse to understand each other. There is also the possibility that both sides could be partially correct and partially incorrect simultaneously and only a person with full and complete knowledge that both sides posses can make good determination as to what each side is correct on.
How can someone determine how much information is "necessary" before coming to their final conclusion? The answer... when they have obtained all the information available.