You can definitely put a whole stick of cinnamon in with your coffee grounds or tea leaves. If your cinnamon stick is very tightly coiled, the hot water may not reach the inside of the coil. Split the stick lengthwise so the water can more easily reach the inside of the bark (that way you get flavor out of both sides of the cinnamon bark instead of just one ...
Most people just use ready-ground cinnamon.
It's much cheaper & just stirs in with no chewy, twiggy bits.
Of course, it looks a whole lot better with en entire stick of cinnamon bark sticking out of a tall, elegant glass/mug, but the amount of flavour you're going to get from it is going to be minimal in comparison.
If you're cooking with cinnamon, it ...
Your tap water probably contains more minerals and your household filter does not remove them completely. (maybe your household filter is not working properly).
Bottled water use industrial food grade filtering techniques, they work better than home filters.
You could ask your city about the mineral content of your water, it should be