Instead of rolling it all into two arrows, expand voting into three categories:
Like | Quality | On-Topic
Dislike | Bad Quality | Off-Topic
The reasoning for this is that on other sites like Reddit and Voat, users tend to use the votes as an "I agree" or "I disagree" button, despite the efforts of the admins to convince them otherwise. This stems from too much ambiguity in what the votes represent - and with a tag voting system such as this, those ambiguities could be eliminated, causing people to be much more critical with their votes. Users would get one vote for each category - for example if there was a quality shitpost in /s/Jontron, I would put votes into
.>Like< | >Quality< | On-Topic
Dislike | Bad Quality | >Off-Topic<
(Even though we all know that any internet page associated with Jontron is automatically a shitpost sub) Perhaps a "shitpost" tag could be rolled into "Off-Topic" to further reduce voting ambiguities.
The idea is not without merit.
But I'm not sure how effective it would be. You seem to be assuming that the bulk of the misused votes are due to confusion, or not understanding the correct purpose.
You have more faith in people than I do.
I think most misused downvotes come from people that know exactly what they are doing. To them this would just allow them to downvote things they don't like/agree with three times.
Also, I think not enough people vote as it is. Making things more involved would probably end up with less people voting.
I agree that there are flaws in the current up/down system. But I'm not sure this would be the answer.
I think Slashdot used to have something similar to what you propose. And it seemed to work for them, but they didn't allow everyone to vote. Only known users. That makes a big difference.