It's my favourite puzzle about meditation. You have to approach it sideways, but if you really try that it won't work. So you try hard not to that instead, which won't work either. It's kind of funny how focusing on your breath tends to work, but... don't hold that thought.
It's a contradiction, and that's okay. Get comfortable with it ;-)
None of those things are doing meditation wrong, they're just things minds do. The mistake is in focusing on them instead of focusing on the process generating the changes.
The approach that finally worked for me (and one I still use today, years later) I learned from a book by Adrian Wells called Metacognitive Therapy. It's not self-help, actually; it's a clinical manual for therapists, but I read it anyway. The idea, in my own words, is not to engage with the thought but simply to recognize it, almost like greeting it and saying goodbye in the same breath. A thought comes, and you say, "Hmm, okay, I just thought that. That was a thought. That's fine." And you move on.
Of course, the book goes much deeper than that, but this simple approach helped me enormously with rumination and anxiety around intrusive thoughts.
It became clear to me at some point that meditation was another way I was creating some standard for good or bad, and comparing myself to it. Would be annoyed at myself if I weren’t “mindful enough” etc. I think even if you don’t have this tendency, but you’re still meditating with the intent to be in a certain state or arrange things a certain way, you’re still doing a minor version of this - how I was before, not good, how I am now, better.
What I do now instead of meditating is to just sit without doing anything on purpose. I don’t try to prevent myself from thinking or direct my intention in any way. So I just am as I am - maybe my mind wanders, maybe it doesn’t. This way there’s no contradiction. When I’m anxious, I’m anxious. When my mind wanders, it wanders. When I think about cheese, I’m thinking about cheese. If I think “I’m feeling kind of present right now”, then there’s “I’m feeling kind of present right now”. And before I know it, it’s something different
Then don't. It's exactly the point to not do that. Meditation needs non-attachment imho. Because when you're attached (I need this to feel a certain way, something must happen) you can't really be observant of your thoughts and sensations, you're one with them.
All that said I think if someone wants to choose to meditate, or to spend time being aware of feelings or whatever form of meditation they choose, they’re totally welcome to do that. And I agree that if you do that, it’s helpful to focus on the process rather than the outcome.
A side note: the experience of meditation alone and in groups is very different. There’s a reason why people who are “serious about it” (or seriously interested) do it in community.
A common occurrence is the thought that oneself is thinking or labeling too much. This is close to the recursion / self-reference you are referring to. The practice thereof is to simply accept that (and your labeling of inner activity and all wrapped up in that) and carry on abiding. The inner chatter that you are not meditating well enough is a funny one.
Liken it to brushing your teeth: you can't really judge it. You probably did it good enough. It's more important to keep doing it steadily every day than to evaluate progress.
Meditation, in the Therevada tradition at least, never claimed to be nothing, it uses doing to undermine it.