Throbert,
ученик русского языка and учащийся русского языкаwould sound awkard to a Russian speaker. So would also ученик карате and ученик французской кухни. The word student in English has a broader meaning, but ученик is rather confined in Russian.
ученик (in school environment) is the equivalent of grammar school and high school student in USA. Now, technically, a grammar school student is ученик младших классов, and a high school student is ученик старших классов (can be called старшеклассник). As soon as you enter a what you would call 'community college', technical school, or college or university (which is also somewhat confusing for Russians as колледж is a lower rung, while proper higher education in Russia implies graduating from университет or институт) you become a студент. So we, basically, have ученик (grades 1 to 11) and студент (any school or college after high school).
I am student of Russian should be translated as Я учу русский язык (liter., I learn Russian) or Я изучаю русский язык (liter., I study Russian).
I am a karate student - Я занимаюсь карате. (99% would say that)
Instead of я -- ученик французской кухни you can say Я изучаю французскую кухню.
CoffeeCup, let me further explain the usage of ученик that you talk about. Let's consider this example: Матисс - ученик Моро. In the Russian language this phrase means that painter Henri Matisse is not only Gustave Moreau's student, but his follower, that Matisse took something from Moreau in terms of style, expression, artistic rigor, etc. Similarly, a karate master could say of his student something like: "Он - ученик моей школы" (He is the follower/disciple of my karate school).