Does Latin have words for the various types of diets, e.g., "vegetarian," "vegan," etc.?
Not really. There is certainly plenty of discussion of diets in Latin literature, whether for health or religious purposes, or just for anthropological interest. But there was no single name for any particular type of diet. A good example of this is vegetarianism. Vegetarianism was known and even practised. Yet, in discussions of it, it is always described rather than labelled.
Seneca, following Pythagoras' example, was a vegetarian and discusses not only the health benefits of such a diet but also the philosophical reasons behind it. Below are two typical references to vegetarianism:
Sotion dicebat, quare ille animalibus abstinuisset …
Sotion used to say why Pythagoras abstained from animals (or living
Seneca the Younger, Epistles, 108.17
Alienigena tum sacra movebantur, sed inter argumenta superstitionis
ponebatur quorundam animalium abstinentia …
Some foreign rites were at that time being inaugurated, and among the
proofs of sanctity, the abstinence from certain kinds of
animals/living things was ordained …
Seneca, Epistles, 108.22
His vegetarianism is described in terms of what he doesn't eat (which I've translated as living things since Seneca and Pythagoras' concerns about meat are that they may be consuming others' souls residing in animals). Nowhere in this discussion, however, does a noun for 'vegetarian' or 'vegetarianism' occur.
Here, Tacitus speaks of Seneca, but in terms of what he does eat, using victus:
Seneca … dum persimplici victu et agrestibus pomis … vitam tolerat
Seneca … who sustained life on an extremely simple diet of fruits from
Tacitus, Annals, 15.45
This use of victus to describe a particular diet is common.
Fennis mira feritas, foeda paupertas: non arma, non equi, non penates;
victui herba, vestitui pelles, cubile humus
The Fenni live in astonishing barbarism and disgusting misery: no
arms, no horses, no household; wild plants for their food, skins for
their clothing, the ground for their beds
Tacitus, Germania, 46
Agriculturae non student, maiorque pars eorum victus in lacte, caseo,
For agriculture they [the Germans] have no zeal, and the greater part
of their diet consists of milk, cheese, and meat
Caesar, The Gallic War, 6.22
Much later, and using 'vegetable' ([h]olus) specifically, Jerome wrote:
nihilque ita scias conducere Christianis adulescentibus ut esum
You should know that nothing is so good for young Christians as a diet
Jerome, Letters, 54.10
As you can see, the authors are describing vegetarianism (and other types of diet) but clearly they don’t have a convenient one-word label to apply.
The closest I could find to such a label is in Pliny. However, these words don’t seem to occur elsewhere, and they are describing far-flung (sometimes mythical) people. To that end, I wonder if their Greek etymology was specifically chosen to sound different, exotic, fantastical or perhaps to give a scientific gloss to them, like he’s reporting from field notes.
Ichthyophagos omnes Alexander vetuit piscibus vivere
Alexander made an order forbidding a fish diet to all the Fish-eaters
Pliny the Elder, Natural History, 6.95
Agriophagi pantherarum leonumque maxime carnibus viventes, Pamphagi
omnia mandentes, Anthropophagi humana carne vescentes
the Wild-beast-eaters, who live chiefly on the flesh of panthers and
lions; the Everything-eaters, who devour everything; the Man-eaters,
whose diet is human flesh
Pliny, Natural History, 6.195