Bath oils are primarily just that: oils derived from all sorts of sources, including sunflower, almond, coconut, and jojoba—and from a virtual plethora of plants and flowers. Some bath oils contain volatile (fragrant) oils that can potentially cause allergic reactions. Other bath oils are formulated with slip agents (ingredients that help the oils move over the skin) as well as with mineral oil (mineral oil can be more soothing for the skin than plant oils because it poses minimal to no risk of irritation). Some even contain water-binding agents, which are hardly necessary since the skin will be water-laden whether they are present or not, and because their effect is washed down the drain.
If there is any distinction between oils, it has more to do with how greasy they feel and how much irritation they can cause than with any healing benefits they may have. Plain mineral oil can be an excellent bath oil because it is fragrance-free, gentle and emollient, and unlikely to cause irritation or breakouts. Safflower, sesame, almond, avocado, and even olive oil can add the slip and emollience needed by dry skin because they stay on the skin and are not rinsed down the drain before you step out of the bath or shower.