The bacteria you want grow in the filter. There will be a very small load of them in the water, but they only thrive when anchored on surfaces. High flow areas like inside your filter provide them maximum nutrients and oxygen, and so grow the largest amount of them. There's a lot of variations (plastic bioballs, ceramic rings, sponges, fiber, etc), but all biological filter media is very high surface area to give the bacteria lots of space to anchor and grow.
There's one exception to this. It's common in a cycling tank (or an existing tank if the water quality deteriorates) to have a very high load of bacteria in the water. This is called a bacterial bloom, and it's not actually the "right" bacteria. They do eat ammonia, but they do so very inefficiently.
The water itself is only what drives the cycle, not what cycles (there used to be a lot of noise about "old water" and this myth persists today in recommendations of using water from mature tanks to supposedly cycle new ones), but it's been soundly debunked.
The daily water changes aren't so much "if you like" - if you cycle with fish, it's the best way to ensure the fish remain healthy (or evne alive) through the cycle process. Fishless cycling doesn't involve regular water changes.