Most of the terms you mention are not mutually exclusive.
Commodity is not a specialized term and it just denotes some (mostly) fungible economic good (see here or here). So water is a commodity.
Natural resources can be defined as (OECD 2005):
Natural resources are natural assets (raw materials) occurring in nature that can be used for economic production or consumption.
So water is also a natural resource.
In terms of economic classification water (tap or bottled) is private good because it is both rival and excludable (see Mankiw Principles of Economics pp 226). However, here it also depends which water we are talking about. I assume you mean tap or bottled drinking water. An argument could be that water in rivers, lakes and seas is non-excludable (in the case of lakes it would also depend on size of the lake). If we would be talking about water in a sea it would common resource because of non-excludability.