“We know the girl already 2 patch, We love each other and are going to get married. Recently, however, there has been a small argument between us, she considers, that since we are planning a wedding, then I should go to her parents and formally ask for her hand. It seems artificial and unnecessary to me. After all, her parents know about our plans anyway, so why this whole nativity scene? Besides, I don't really know, what such proposals should look like; Should I e.g.. kneel in front of her parents, buy her a ring, that the tradition would be done?”
Jej prośba wskazuje na to, that her family values old traditions. That's why I think, that you should, however, overcome your resistance and propose in the traditional way. Gdybyś ograniczył się tylko do suchego poinformowania rodziców dziewczyny o swoich zamiarach, they would probably consider this form of marriage proposal not solemn enough and not very elegant.
Of course, you don't have to kneel in front of anyone. That's enough, if at the appropriate moment you say e.g.; “You probably already know, that we both love each other and that we have decided to get married. W związku z tym chciałbym państwa prosić o rękę córki oraz o przychylność dla naszych planów”. The family should be notified in advance about your visit, and you should show up in formal wear, with a bouquet of flowers for the future mother-in-law.
I think, that it would be nice for her, if she got an engagement ring from you. It can be very modest, e.g.. silver, but it should be. It's a nice habit and worth cultivating. If you would like to make one more bow to tradition, put her ring on the ring finger of her left hand and ask her, to clench her hand. Superstitious people say, that it guarantees happiness and durability of the relationship, which I sincerely wish you.