Actually, in both passages, Jesus is speaking to specific people (the first passage is a multitude, but they are still a specific group of people...since Jesus was not speaking the words directly to 21st century Christians).
Both texts can be applied to all mankind, however.
With the "judge not" passage, it is really referring to matters of the heart...for example, it is not my place to condemn someone to hell. I can be an example of Christ to people, but I cannot sentence them to hell. For the "judge with a righteous judgement", it is more about not judging people on outside appearances, but rather basing your opinions on what you actually know. For example, meeting someone who dresses like a "thug" doesn't automatically make them a thug. Does that make sense?
With a husband in seminary, I believe that there is only one correct interpretation...that's God's interpretation. And I say that because people take can one verse and decide what it means for whatever reason they wish to bend it for. Problem is that the are not looking at it from the context it is written...or even in the languages they were written in (which in Greek & Hebrew, the way it's written gives a BIG indication of how the verses are to be interpreted...and trust me, we're all about the languages here as DH is studying on them like crazy

)
However, I also believe that there are multiple
applications for verses, which seems like that's what you mean more than interpretation. To give a simplified explanation...let's say a verse was written in Hebrew to say "sit in the chair." It can only be interpreted as "sit in the chair." But the application might be different. Is it sit in the chair with your legs crossed? Sit in the chair with one leg tucked under? Etc? Well, all of them can be equally fine...after all, you're still sitting in the chair.
But anyway, I think what I just said it really more of a long explanation...but it seems like you and I agree that both texts can be applied in different situations. I just think that
both sides of the fence should be careful before throwing around verses. The Bible is not meant as a weapon. It is a Book proclaiming the gift Christ is giving to mankind. It is not a Book meant to be used to force people to God. Yes, I believe Christians should stand up for what they believe in, just like anyone else...but they should also remember that God is not a dictator. He wants people to
choose Him, not for them to be forced or threatened into Christianity.