These are not terms with a clear meaning in England. We just have "secondary" schools - there are no sub-types. I don't know much about the US system but I have only ever heard of "high schools". If you are going to describe a noun like "high school", then it is good to use a relative clause e.g. a high school which teaches students to a high standard.

The language in the U.S. would be " A high school level education", which simply means that you have graduated high school but have not ever attended a college or university. This is very common terminology in the U.S. and you will find this question on almost any kind of application for employment, benefits, or medical care. Really, on just about any kind of form.