"...does a pilot typically carry a water container/supply?..."

When I was in the USAF, what I typically saw was AF and Army pilots carrying those little OD green "hip flask" bottles in a leg pocket, and many had a condom or two hidden someplace on them as an emergency canteen (or maybe they hoped to find some real friendly natives???), Navy pilots carried in their chest vest baby bottles of water, with just the flat cap on top. Macho it ain't but they fit. And some seemed to carry nothing...