I think in this case it has a flavour of "to have" more than "to be". Because in this case the meaning is not "The students are here" but "There are students here" which is slightly different.
A tip to understand this is to compare with Spanish: "Hay estudiantes aquí" - literally, "Have students here". That is exactly what we do in Ukrainian: "є" also means "to have" (e.g. У мене є студент = I have a student), so "Тут є студенти" is, basically, "Here have students" which means "there are" :)
Тут студенти = Студенти тут = "students here" --> "The students are here" (in English you don't say "Students are here", right?) --> yes, you are right, the students in this sentence are specific
Тут є студенти = "There have students" --> "There are students here" (in English you don't say "There are the students here", right?) --> yes, not specific students, just some/any students
You are completely right, nice observation! :)
Even if there is a linguistic term, don't use them... Those are made for people who study linguistics so that they can communicate faster. For us the terms don't make communication faster because we need to spend extra time understanding them, so why would one use them in that case... The words that you used like "specific" and "any" are perfect for the explanation!..