On the specific case my impression is that you would be voted down simply because you're disagreeing with the OP, right or wrong. No logical argument, evidence or proof of your case would convince them. I think we've all experienced these cases and no system will avoid them, IMO.
Taking the general issues not the specific case, I think the problem with separate "dislike" and "disagree" votes is threefold.
Firstly it requires implementing two vote systems in the same UI. This would lead to confusion, IMO, as people, particularly newer members or members who vote infrequently, will simply mistakenly use one when they should be using the other one. Two similar functions is always asking for trouble in user interfaces in my experience.
Secondly it allows people to both vote on popularity ("Do I like this member") and accuracy/usefulness ("This answer is useful"). It's potentially possible with both voting systems active to vote up an answer for being useful while voting it down because you don't like the member. You could double whammy someone by voting the answer down and the member down. At the moment we just vote on usefulness - not necessarily accuracy, but whether it's a useful answer. Sometimes an answer which provides e.g. a simplified explanation or intuition is better than a strictly accurate answer. I think that's a good overall balance.
Thirdly it makes a member's popularity a legitimate factor in voting. I think this is fundamentally the opposite of what SE tries to achieve. We want people to vote on how useful an answer is, in their opinion. Now I am sure some people use it as a popularity vote if they don't like someone (or just don't like the answer, accurate or not), but notionally at least it's about the usefulness of answers and my impression is that most people seem to vote with their heads and not their hearts.
IMO this is important as the trend is for a lot of questions to be based on theory, as if theory dictates the observations instead of modeling it.
I'd like to see answers linking theory to experimental knowledge more, but I don't think an extra vote mechanism will do it. Getting people to make answers like this requires a lot of "Leading from the front". That is to say it requires the more knowledgeable members and those who post and vote often to both write answers like these (e.g. "..and experiment X shows this...") and to encourage people to expand their answers by using comments (e.g. "I'd vote up if you could add some intuitive or experimental/empirical support for your answer, which is otherwise good").
I don't think there is a shortcut to achieving the goal you want. It requires community effort.