swaps for/with

Yes.

I swap this for that

I swap it with you.

So it's incorrect to say "I saw this bottle with that bottle to solve the puzzle."?

I would swap bottle B for bottle A.

If you have bottle A and I have bottle B then I would swap with you.

Hamilton swaps road for/with water.

What does this mean? Doees it mean that a man called Hamilton stops using roads and starts to use water as a means of getting from one place to another? If so, then he swaps the road for water.

It's a title of a clip of Hamilton going sailing.