I never really understood the 'stopping' part, how is that even possible? At best you can simply refocus it to something you find peaceful, if that is the case how is meditation any different from say watching a movie that you love or playing drums.
How interesting that that thought wandered by just after this other one. It's "smile", that one feels pretty good.
This next one is "boredom"... Fascinating! What is that about? Why do I feel it? What happens if I use "smile" on it?
What about this monologue? Can I step back and observe those thoughts?
The Zen master says he's thinking about nothing while meditating. That doesn't mean no thought, it means observing the thing that Null pointers point to.
See stopping as a metaphor. You are not "not thinking" ever but you can ignore some thoughts. And we do that all the time - usually thoughts we don't like. With meditation you'll learn to do that a bit better. A thought pops up in your head and instead of following it and expanding on it, you simply let it slide. Another one pops up and you do the same. When I meditate for 15mins at most 4-5 mins of it would be close to "peace" - that's all I can manage. When I'm doing it regularly, (it could very well be placebo) I find myself to be less agitated and not following every thought that comes to my head through out the day. I learn to say "not now" and focus on what I'm working on instead.
The mystical philosophies around meditation describe it with all sorts of features and properties, but I'll skip getting into that here. I'll just mention martial arts in particular. They practice relentlessly so that the actions are instinctive, then they use meditation so that higher-brain noise doesn't get in the way of that hyper-awareness and instinctive action.
With meditation, if you get good at it, you can get lost in a kind of haze that's way less focused than playing drums or watching a movie. Sort of like sleeping while awake.
So that's a lot like "stopping". Sure, neurons will fire, etc, so you're not stopping anything completely, but they also do that when you are sleeping, or when you're completely wasted on weed.
The main difference, I've found, is temporal. By focusing on the breath, you stay grounded in the present moment. Watching a movie will trigger memories or thoughts about the future, which, to me, seem to tax the mind a lot more than simple, objective thoughts about the present moment.
Playing the drums could be like this, but you have to get lost in it and be improvising or sight reading something you've never played before. If you're worried about making mistakes, remembering the music or anything along those lines, you're not as in the moment as you could be. Most people don't play that way.
With practice.