These diets

Evaluated Diets

Vegetarianism

Rules are defined in The European Vegetarian Label:

Ovo-vegetarian - contains egg
Lacto-Vegetarian - contains milk
Ovo-Lacto-Vegetarian - contains egg and/or milk
Vegan - no animal product

Strangely these rules do not exclude, but include. However the basic, unspoken rule, in these labels is that nothing should come from dead animals. Interestingly they note that these rules are part of Swiss law. The European Vegetarian has defined detailed rules to be included. Interestingly these rules also specify animal welfare (no battery eggs) and non-gmo. Also it talks about products used during the product creating, which are not to be found on the productlabel as ingredient.
There is also a labelling system in India (using green and red).

Raw veganism

Raw veganism follows veganism and raw foodism It is a compound diet.

Raw foodism

Raw foodism implies that food cooked at a temperature above 48 °C should be excluded. This rule is not based on the ingredients used, but on how the product is made. Thus, this can not be extracted from the productlabel.

Fruitarianism

Fruitarianism involves the practice of following a diet that includes only or primarily fruits in the botanical sense, and possibly nuts and seeds, without animal products. It seems this a rule covered by ingredient groups (fruits, nuts, seeds).