Two definitions of $H^1(\partial\Omega)$, one using charts and one use tangential gradients

The two definitions are the same. For a reference see page 294 in Dziuk, Gerhard and Elliott, Charles M.. (2013) Finite element methods for surface PDEs. Acta Numerica, Vol.22 http://wrap.warwick.ac.uk/53966/1/WRAP_Elliott_DziEll13a.pdf

I'm not certain one can give an exact answer as to why the first definition is more popular. My personal opinion is that the first definition is a useful tool for showing many results about surface calculus and partial differential equations posed on surfaces. However, in the above reference, the authors manage to show many important results using the second formulation.