En México lo común es decir "botella de agua" y está avalado por la RAE.

if you google it you will see things like:

botella de agua (seen most often, emphasis on the contents of the bottle)

botella con agua (I think these 2 shift emphasis to the type of bottle)
botella para agua

agua embotellada

botella del agua

Is bottled water "botella de agua" or "botella con agua"? "de" would seem like the bottle was made of water or originated in the water.

Probably you never thought that "bottle of water", in English, could be interpreted the same way. (a bottle made of water)

What it really means, both in Spanish and English, is that the bottle was originally made to be filled with water.

But "con" seems awkward.

"botella con agua" is equivalent to "a bottle with water". That is, a generic bottle filled with water. It doesn't apply to bottled water but it can be used in some contexts.

"Botella de agua" is perfect; ignore anyone who says otherwise. What's wrong with "de"? Don't you say "a bottle of water" in English? Does that mean that the bottle is made out of water? Of course not! Prepositions have many uses, and indicating what a material is made of is just one of many meanings that "de" has.

Look at the dictionary of the RAE:

de
5. prep. U. para señalar lo contenido en algo. Un vaso de agua. Un plato de asado.
Real Academia Española © Todos los derechos reservados

a glass of wine - un vaso/ una copa de vino
a box of crackers/biscuits - una caja de bizcochos
a can of soup - una lata de sopa

All measures (indicating quantity), albeit less precise than quart/liter/pound/kilo.