This seems to quite clearly show that the Buddha both advocated and employed the practice of identifying, judging, labeling and separating thoughts.
Rob this quote is about reflecting. Its not about meditation. He is cultivating mindfulness of thoughts. He is not labelling them etc. He is simply saying he is aware of this or that nature of them. He is not putting words on them.
To say you "know" something is not to say you labelled it or named it.
He talks elsewhere about the theory of experiences being pleasant, unpleasant or neutral, but i am not sure if in meditation he is saying to make a thing of labelling them.
I think simple awareness of whether something is pleasant or unpleasant is easy enough. You see labelling means consciously putting a word on things. Yes some teachers say to to do this. I think this could be helpful for some people who are really out of touch with their feelings. I think it could be helpful for beginners but ultimately i don't think its what we are trying to do.
Maybe do it when you are not in sitting meditation. Try to become more aware and mindful when you are doing other activities and try noticing or labelling them then. But in sitting meditation itself, try to keep it pure (free of words and thinking and thought and concepts and such), just come back to the breath if you mind wanders or come back to pure awareness of sensations etc. But let all your thoughts go.
I think this passages sounds like awareness of thoughts. He is not thinking about his thoughts, he says he observed a thought and he was able to let it go.
He is not doing the sort of meditation that we are trying to learn. He is talking about reflecting which is a process of thinking also. So i would say, put this aside until you have grasped the more basic skills.
It in this method of reflecting, he is cultivating awareness of thoughts.
Keep it separate from your meditation. If you want to practice this, keep it separate from your sitting meditation.
But back to your other points that it is contentious, well again if you were working with a wise teacher, he/she would be able to discern whether this technique of labelling would be useful for you but ultimately you would be instructed to drop it. Those against it are probably trying to make sure you keep yoru mediation pure.
You make your own decision but for now don't do it in your sitting meditation.