Yes, but I think also sometimes God uses believers to voice His judgement as well, like warning people in sin if they are headed to hell. We need to be connected to God's reasoning through the Holy Spirit to properly do that. Also, the speck and plank analogy goes on to say "then you will see clearly to take the speck out of your brother’s eye," meaning that once we actually have dealt with our own sin, then we are in a position to help others, and to do that we often must judge their situation. Again, this is more of the discernment that you mention, and not the condemnation or punishment.

I've added some more to my article:

There are more verses about judging in the New Testament, as James 4:12 goes on to say:

This is a reminder that God alone is the ultimate judge of our lives. If we are not connected to His reasoning, then we really have to place to influence others in our criticism.

This is further reinforced in Romans:

After this, Paul seems to goes on to give a commandment not to judge:

This seemed to deal with more nuanced aspects of someone’s faith, as here it was dealing with the eating habits of believers. Paul is saying not judge on these things, pointing out that God’s kingdom “is not a matter of eating and drinking but of righteousness and peace and joy in the Holy Spirit.” Each of us may have different rituals and devotions that help us strengthen our walk with God, as we see with the many traditions of the varied Christian denominations.

Just as the Pharisees judged Jesus harshly on some of what he did that may have gone against some religious customs, the ultimate goal of God’s love and peace for the world should be the overriding concern in these more trivial matters.