According to the dictionary of Fasmer the word, Bogatyrs, was an early borrowing from the Turkic languages and not formed from bogat (rich) or bog (God). He names, each, a cognate in Polish and in Bulgarian.

You said «Bogatur», not «Bogatyr». That's how it's spoken in Mongolian. There the meaning is «brave».

For their place in Russian folklore there is a Vladimir Propp book on «Russian Epic Song». The «place» of various character types are his exact subject. In the actual folk epics there doesn't seem to be a fixed meaning. For example, here, Ilya Muromets is called a "cossack". It's his horse who is "a big bogatyr". If I had to translate I would settle for "beefcake".

edit: the plot is he finds a stone with directions. «Go on the first way and you'll find death. Go on the second way and you'll lose your horse. Go on the third way and you will get married.» Because he is old he settles for the first road and is surrounded by robbers. He rips on oak out of the soil and beats them up.

That's how bylinas are sung.