The like button to communicates two main ideas: one, that you like the content of the post and/or two, that you read the content of the post.
All too often, those two are not the same.
As such, I think it would be a very useful idea to have a "read" button (maybe the "eyeball" visibility symbol) to indicate that you have read the post but don't necessarily agree with it.
You would not need to push both buttons since the "like" button would clearly indicate that you liked the content and read it while the "read" button would only indicate that you've read it but don't necessarily like the post or carry any opinion for it.
I find that, especially when I'm the OP, I want to give people an indication that their point of view has been read, has been processed, and to communicate that fact without necessarily agreeing with them.
This would not be a downvote button and thus an invitation for being negative about a post. As I see it, it would be the opposite... it would be a way to increase respectful discourse in so much as you acknowledge you read their post, respect what they have to say, even if you don't necessarily agree with what they have to say.
I'm not sure about this. It has some appeal to have an "engagement score" that goes both ways (how much you engage others, as well as how much you engage). But due to the stateless nature of web pages, that's harder to do (with any accuracy at least) than you'd think. Since AFAIK this isn't a for-profit site, they probably have to be careful how much server resource consumption and code maintenance each feature suggestion causes.
I agree that this would be a good addition. It may or may not increase civil discourse, BUT, it would allow posters to see how engaging or interesting their writing style is. It could provide readers with a way of perhaps showing that they are left at least neutral but maybe even thoughtful about the post, when they perhaps were to scared or passionate enough about it to post a comment. I like the idea.
Why limit the choices? How about "I don't care", perhaps "I read the first paragraph and turned off!" or my favorite. "You're an idiot!" (The last is not directed at the author, It's just me being stupid. Again.)