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).

Sattvic diet

The Sattvic diet is a diet based on foods with sattva quality. Sattvic diet places emphasis on lacto-vegetarian ingredients that is fresh, seasonal, and naturally sourced (non-processed). Grains, ripe vegetables (not unripe), fruits, nuts, seeds, legumes that is well prepared and freshly cooked are favored. Oils and spices are considered sattvic, if naturally sourced and used in moderation. Milk and milk products are emphasized in many forms.
The rules seem to be mainly based on ingredient groups (grains, vegetables, fruits, nuts, seeds, legumes. However this definition has some attributes that are different:

It should be non-processed, which includes many products.
Ripe vegetables, which seems not very relevant in the OFF context?

Buddhist vegetarianism

The wikipedia article on Buddhist vegetarianism is a bit confusing as it seem to relate more to theory. There are however a few common practises mentioned:

Meat - the seem to talk specifically about meat, thus inclusing poultry and fish? (ingredient group)
Fetid vegetables - traditionally garlic, Allium chinense, asafoetida, shallot, and Allium victorialis (victory onion or mountain leek), although in modern times this rule is often interpreted to include other vegetables of the onion genus, as well as coriander. (ingredient group)

Jain vegetarianism

The Jain vegetarianism cuisine is completely vegetarian and also excludes onions, potatoes, brinjals and garlic. Only a few ingredient groups are added to the vegetarian diet rule.

Macrobiotic diet

A macrobiotic diet (or macrobiotics), is a dietary regimen which involves eating grains as a staple food, supplemented with other foods such as local vegetables, and avoiding the use of highly processed or refined foods and most animal products. The definition is a bit vague and seems to have many exceptions and detailed rules. Again some observations:

local vegetables - is local an attribute that can be supported by Open Food Facts.
highly processed or refined - is this a subset of processed?
most animal products - can this be covered by an ingredient category?

Kangatarianism

Kangatarianism is a recent practice of following a diet which excludes meat except kangaroo. (ingredient category:kangaroo meat).