You may already be familiar with this (in which case, my apologies), but the general term for that sort of non-linear text is “choice-based IF”, and Twine (including Sugarcube, as Manon mentioned), Ink, ChoiceScript, and Inklewriter are all different ways of writing it. (Plus some more besides.)
Ink is my favorite, but Twine is the most versatile, ChoiceScript the most popular, and Inklewriter the easiest to get started with.
Oh, yes, I am familiar with IF, but I thought Ink was a special branch. So it is just a programming tool, right? I am only familiar with Harlowe, and I will probably do with it for some time. Sorry for this misunderstanding. I thought the style when you don’t give question choices but offer, say, a variety of adjectives inside your narrative, has a genre name to it.
Ink is a scripting language for choice-based IF. The main advantages it offers over other systems (IMO) are a more minimal/writing-friendly syntax, and the ability to easily integrate it with a full game engine or programming language (so you can, for example, use it to handle the dialogue in a graphical, non-IF game).
I thought the style when you don’t give question choices but offer, say, a variety of adjectives inside your narrative, has a genre name to it.
If you’re talking about having clickable links inside the text, like you would make with Twine/Harlowe, then I think the usual genre name is “hypertext fiction.”
That might sound like the ansqer I was looking for. So when I use active parts of text instead of direct questions and choices in the end of given page, the subgenre name would be hypertext vs. choice-based fiction? Still, hypertext sounds like more general term and actice parts of text instead of questions and fully formulated choices are both a kind of choice you make 
Perhaps I sm looking for something which doesntneed naming and is jurt a matter of format/style.
I think hypertext fiction is generally considered a subset of choice-based IF. You’re right that from a structural design perspective they’re not that different & it’s mostly just a question of presentation. For example, Ink would typically be considered to be a tool for choice-based IF, but you can actually modify it really easily to add inline links, so you can make hypertext fiction with it as well. And conversely it’s very easy to write Ink-like choice-based IF in Twine if you want to, by just listing all of the links on separate lines below the main text.