There are five major types of bottles that can be used for this project, but the inner bottle that actually touches the water should always be HDPE (Nalgene), which is cloudy semi-hard plastic. Soda bottles are made of PET. When PET gets hot, it may leach DEHP, a potential carcinogen.
Containers made of lexan, the clear plastic that is often brightly colored, may contain traces of BPA which may interfere with your hormones. (It is an estrogen-like compound. Doesn't sound good to me.) Then there are PVC (polyvinyl chloride) and polystyrene bottles, which can also produce carcinogens. Polypropylene or LDPE may be safe choices.
The photo shows the alternate outer bottle - a 1 liter PET soda bottle with the bottom cut off about 2 inches from the bottom and attached on just one side with duct tape. This allows you to hinge the bottom to load the black-painted HDPE bottle.