Is sparsity induced penalty in source separation "Entrywise matrix norms"?
I haven't read the paper, but from the information you provided it appears the appropriate definition is the first one you gave, i.e., the "entrywise" matrix norm.

The text you quoted refers to the norms being "elementwise" which is equivalent to being "entrywise". One way to look at this norm is as the $l_1$ equivalent of the Frobenius norm. Essentially, you are replacing the sum of the squares of the matrix elements with the sum of the absolute value of the elements.

Trying to  minimize the $||H_g||_1 $ norm using this definition, is an approximation for trying to reduce the number of non-zero entries in the matrix $H_g$. You would have to examine the problem formulation in the paper to see if that makes sense.