流れる and 流す: water flowing and music playing
Compare water flowing with making it flow, and examine the meanings involving music and tears.
The course and direction of water
川の水が海へ流れています。 says that river water is flowing toward the sea. With 流れる, read ながれる, the water is the subject and へ marks direction.
雨水が道路を流れていました。 says that rainwater was flowing along the road. 道路を marks its course; no person causing the flow is identified.
Wash something away with water
水を流して、洗面台をすすぎました。 says that someone ran water and rinsed the basin. 流す, read ながす, focuses on causing the water to flow.
泡を水で流してください。 asks someone to rinse away the foam with water. 泡を is what is removed, and 水で is the means.
Music and tears
店内には静かな音楽が流れています。 says that quiet music is playing in the shop. It describes the audible background rather than identifying who started it.
映画を見て涙を流しました。 says that someone shed tears while watching a film. 流す does not imply deliberately producing tears.
Does 涙を流しました imply that the tears were deliberate?
Show the answer and why
No. It reports shedding tears; whether the crying was deliberate requires more context.