Yes, in Spain, ‘instituto’ is the most common term for what in the U.S.A. is called a “high school”. Just to confuse you further, the word ‘colegio’ means “primary school” (including “elementary school” and “high school”), not “secondary school” — the opposite of the English word “college”.
It seems like that, but I think not. I see on web searches that you can say "el instituto" or "instituto" for high school, same as in English you might say "I am going to the high school" (referring to the actual physical buildings) or "I am going to high school" (here I am talking about my level of learning)...
Yes, ‘Soy un estudiante del instituto.’ would mean “I'm a student of the high school.”
With very few exceptions, Spanish doesn't have compound words composed of two bare nouns; instead, Spanish uses the preposition ‘de’. So you can't say *‘estudiante instituto’; you have to say ‘estudiante de instituto’.
The clues beneath the word puerto say "institute and college." It corrected me to "high school", saying corrected answer is" I am a student in the high school." So the next time it came around, that's what I put. Then it corrected me to I am a high school student. When does it make up it's mind? I find this very frustration, and difficult to know what exactly to memorize, as it seems very changeable based on the 'mood' of the algorhythm. Bleh!
In England there are no 'High Schools' they are called Colleges so why is my answer not accepted. I asked to learn Spanish with English not with American. At least you should accept both versions English and American. This is the worst course I have ever done in my life. Full of mistakes. I thought the Crown level is better but it is not! Just as rubbish as the rest. I would not pay a penny to do it.