Meditation helps to calm the nervous system. Counting to 10 (1 count per inhale + exhale), or simply focusing on each inhale & exhale, or repeating a word or multi-word mantra, is what I do personally.
It's especially nice in a Sauna with some mystical atmospheric music (such as Anugama, one of my favorites for meditation).
I have found that it improves my ability to concentrate (especially useful when studying or working) and to calm myself in stressful situations.
Not all meditative traditions require belief in a God.
See Hindu atheism for example: https://en.wikipedia.org/wiki/Hindu_atheism
I would also say prayer like you see in Abrahamic religions is its own type of meditation. You also enter a sort of meditative state in sports where you kind of feel in the moment and "in the flow" where there's not God involved... at least not consciously.
I suppose that would be less guess and more...belief. :)
Meditation is not just about accepting emotions and thoughts; you can get such by talking with a close friend or visiting a mental health counselor. Many people claim to receive new insight or guidance amidst the mental stillness. Two common explanations is that it's either God or your Higher Self sending a message.
Prayer is absolutely a form of meditation, arguably a more advanced form. If meditation fosters observation of self and others, prayer takes that observation and combines it with will to cause action.
What you're ultimately getting at is the difference between "going through the motions" and proper meditation and prayer. Most spiritual systems discuss a "right" way to meditate or pray which focuses less on the external actions and more on your internal desire.
I’m not aware of an explicit ordering. But the other seven steps complement and deepen the meditation.