Don't. Embrace it...
Let me ask and then answer a question: Why do you CARE that something was downvoted? Because when things are downvoted you lose privileges, comments are hidden/greyed, and you can even be shadow-banned. So downvoting or "disagreement" is problematic because it is PUNISHED. Reddit has an identical issue.
That's a problem with the site's design. People aren't going to stop using the downvote as a disagreement button but yet the site is designed around the concept that it is a moderation tool? That's completely irrational, and only results in group-think.
So what's the solution? Well you can hope that people change and start using the site as you intended, or you can design the site around people as they are rather than as you'd like them to be.
Three buttons:
- I agree
- I disagree
- I believe this is inappropriate/rule breaking/low quality/flag it
Neither the first nor second are moderation actions. They're opinions and will be displayed as such e.g. (+10/-100) for a controversial opinion, and (+100/-10) for a popular one.
With the third option, you have to mitigate abuse. It is not a "super downvote" or a "super disagree." If someone uses it:
- They can either vote OR flag
- They can either reply OR flag
- The flag has a cooldown
- You abuse the flag you lose the flag
As soon as they flag a post that post is dead to them, they cannot interact with it. This stops them from using it in their own little vendetta, and frankly I believe people would prefer to add to the public disagreement score (or reply) than to abuse the private flag function.