I have seen vegetarian translated as both bejitarian and Saishokushugi.
bejitarian is obviously an adaptation of vegetarian.
Saishokushugi can be loosely translated as faith to vegetables.
But what's the difference between the two terms and when would you use one over the other?
菜食主義 (saishoku-shugi) is vegetarianism, and 菜食主義者 (saishoku-shugi-sha) is a vegetarian, a person who practice vegetarianism. To break down:
To me, ベジタリアン and 菜食主義者 are almost identical as far as meaning goes. I believe you can safely use whichever you prefer in most cases. In casual settings, I feel many people prefer ベジタリアン, presumably because it sounds less stiff and is a lot easier to pronounce.
The number of vegetarians in Japan is probably small, and I feel few people in Japan practice vegetarianism for religious or ethical reasons. Most vegetarians in Japan refuse to eat meat for health-related reasons, so 菜食主義者 and ベジタリアン usually refer to such people. I'm vaguely aware that there are many sub-types of vegetarianism in western countries (including veganism), but they are scarcely recognized here in Japan.
I took a look at the Japanese wikipedia article for Vegetarianism. I'm basing my answer on that, because I have little personal experience although I've been asked this a few times.
ベジタリアン is, as you said, an imported word. It has the same meaning as in english. There are different sub-types of vegetarian, etc. It also does not indicate the reason you are vegetarian. It could be for political beliefs, religious beliefs or even because you saw an animal getting killed once and just can't eat meat anymore.
菜食主義{さいしょくしゅぎ} denotes specifically a kind of vegetarianism guided by buddhist practices and beliefs. According to the Wiki article, it (sometimes) has an extra restriction on eating certain kinds of roots. Guided by the Wikipedia article in English on Buddhist Vegetarianism it seems some variations of Buddhism also disencourage consuming vegetables if this kills the plant, thus things like garlic and onion are also forbidden (this is called 禁葷食{きんくんしょく}). But not all buddhists follow this principle; some encourage it, others reinforce it only for monks, some not even for monks. There's a lot of variation apparently.
So when you talk about 菜食主義者{さいしょくしゅぎ}, you talk specifically about the reason one has a dietary restriction. It is because one follows buddhist faith and principles.
The short answer if you're looking for how to talk about your own dietary restrictions is motivation: unless it is this specific Buddhist set of beliefs that guides your vegetarianism, use ベジタリアン (ヴィーガン if that suits you better).
