My take on your standard '3 heart run' is only a restriction on heart pieces/containers (obviously lol), bottled fairies or milk/red & blue potions, and the life upgrade from the Fairy outside Ganon's Castle. Carrying a fairy/healing drinks and reducing damage by half are pretty much the equivalents of having 6 hearts. Also, no Nayru's Love .. using that IMO also defeats the purpose of beating the game with only 3 hearts. Everything else is fair game though, Biggoron Sword included.
Getting / using only what is absolutely needed to beat the game is a more hardcore minimal 3 heart run, which in some people's eyes even means no using the shield except in required battles (Twinrova I believe is the only one that falls in this category) and puzzles. I've done this one before, although I did use the shield.
Whichever way you go, good luck
3 heart challenges are fun.