The best way to distinguish between these terms is to look at what other terms we contrast them with.
'Still water' is used as a contrast to 'sparkling water'. If it has bubbles, it's sparkling or fizzy. If it doesn't, it's 'still'
'Mineral water' refers to water taken directly from a natural spring or source. This is in contrast to 'tap water', which comes from reservoir.
'Fresh water' is not a term used for a commercial product. Fresh water is drinkable, whereas salt water (from the sea) is not. So, for example, you might say that an island is uninhabitable because it has no fresh water.