If you want a free and cozy place in Denpasar, you can go to this river Tukad Bindu in Kesiman Denpasar - Bali
Its very easy to find this place, near from banjar, you can park your motorcycle or car.
No entrance fee / ticket needed
In the morning and afternoon, some people do workout in this place. you can find any four workout tools near the dam.
if you hungry or thirsty, you can buy some foods and drinks at this place
you will find kids playground and a bird cage and many seats to enjoy your day and the river view
I also, really enjoyed the small and important details that you mentioned, such as that the place is accessible by both a car and motorcycle and that it is free.