This is a completely off-topic tangent, but as a fellow non-religious person I'm sorry: Being non-religious and being an atheist are mutually exclusive positions.
Being non-religious means you are apathetic to religion thereof. God(s)? Souls? Afterlife? Commandments? Nope, you don't care about anything concerning religion one way or another.
Being atheist means you believe in no god, no souls, no afterlife, no commandments, and so on. This is, ironically, a form of religion. You care about believing in no religion.
Some people are agnostic, meaning they are religious but don't subscribe to a particular god or doctrine. However no one makes the distinction between atheism and non-religious that you're making here. It may have something to do with the fact that society in the US is so pervasively religious that the only way to escape it is to explicitly identify yourself as an atheist.
Being non-religious means you simply aren't concerned. Are there gods? Great. Are there no gods? Awesome. Jesus is the one God? Okay. Zeus leads his pantheon of gods? Nice.
Atheism meanwhile is a deliberate belief in no religion. It's different from simply not caring about religion at all, because you do care about religion insofar as to not believe in it.
The way I see atheism is that a belief in no religion is, by its nature, a religion. Joe is a christian and believes in Jesus, Bob is an atheist and believes no god; both are merely two sides of the same coin. You can't call yourself non-religious if you believe in a religion, whatever the specific form.
Being non-religious means you don't care; it's not that you don't believe, you simply could not care less one way or another. Wikipedia appears to call it apatheism[1] and more broadly irreligion[2], but semantics aren't the focal issue here.
Agnostic means you believe it's impossible to know whether god exists or not.
I do agree with what you're trying to describe, that many people in some European countries just don't care one iota about religion. I don't think Americans can really understand that without living in a secular society. It's such a non-thing in our lives that, non-religious, agnosticism and atheism, etc. all tend to get mixed into one.
Society has become so secular that those technical definitions have become essentially meaningless. God doesn't exist/I've given it no thought/I don't know if a God exists/I don't acknowledge the existence of supernatural entities all are essentially the same position because it takes up so little of our time or brain power and has so little consequences on our lives.
I stress "some" countries as some European countries, or just small parts of those countries, are still fairly religious.
It sounds like you're conflating meditation-the-exercise with spiritual and religious approaches.
Fundamentally they are unrelated. Taking it to the extreme one can consider the mind to be a process produced by the brain+, no soul involved. Taken that way, meditation is no different than weight lifting. The same way a muscle+ specialises itself depending on training (endurance vs strength vs explosive vs volume) the brain (and thus the mind) also specialises in whatever it gets most exposed to. The same way one can lay out a physical workout plan for a specific desired outcome (including rest), one can lay out a mental workout plan for a specific desired outcome (including rest). The latter is meditation.
Meditation may exist in religious contexts, e.g Buddhist or Zen, but even then many forms are in practice detached from any religious belief, with no koan or mantra. e.g Ānāpānasati (sit, and simply watch the breath) and Sōtō shikantaza (meditation with no objects, anchors, or content, striving to be aware of the stream of thoughts, allowing them to arise and pass away without interference). These are fantastic tools to unlearn bad (sometimes traumatic) mental habits, just like one learns to have smooth but effective muscle action instead of being tense and twitchy and forcing it through.
+ I'm using "brain" as a shortcut for a system that is vastly more complex, just as I use "muscle" for a system that is equally as complex.
These practices date back thousands of years, and likely came out of the understanding of the importance of meditation for a healthy society.
The importance of meditation, yoga-like practice and so on has been muddled in our neo Christian western cultures.
Christianity separates mind, body and soul.
So perhaps now US style Christian prayer has become more about worship and less about self awareness.
That said, weight training completely transformed my relationship with my body. You can only lug a big hunk of iron into the air so many times before you start thinking about how much we really have in common with a gorilla. Yeah we grew opposable thumbs and a much more active prefrontal cortex, but 96% of our DNA is the same as theirs after all. By the pound we're mostly monkeys or something like them and it's a bit conceited to imagine otherwise.
If you're a knowledge worker who sits in a chair and thinks all day it's easy to believe you're nothing like a monkey, but strenuous exercise dispels that notion because it recruits all of your body's monkey systems and makes all of the parts of you work together. In retrospect it shouldn't be surprising at all that letting most of your anatomy wither away is unhealthy and puts you out of balance. It's like getting your car serviced but telling the shop that you only want them to look at the electronics.