What does Graduate school mean: Master, PhD, or both? In the US one seldom reads about Master and much more about Graduate schools. I have the feeling that Graduate school can also mean doing a doctorate/pursuing a PhD degree; thus I am confused with the lack of clear separation here as it isn't a problem with the European distinction of Bachelor, Master, and Doctorate.

Is there a general rule that can be applied to understand what educational level is meant? Like for PhD there is an department admission process required and for Master there isn't, or something like this?

Moreover, one could say to have graduated from High school, from college, from Master's...from kindergarden if you want. Wouldn't it therefore be more logic not to use "graduate" when talking about some specific educational level and instead use terms such as Bachelor, Master, PhD?
Graduate school refer to a school that offers advanced degrees (MS, PhD, or both), even if a school offers only MS degree, it is still called graduate school. I think med-schools are also called graduate schools (since they offer MD and OD degrees).

There will always be some sort of an admission process. Student usually apply for MS program, finish it then join a PhD program. Although in some cases, students can combine MS and PhD together and save 1-2 semesters.