General form of dilations and translations of wavelet function
Yes, they are equivalent, since $j\in\mathbb{Z}$. Note that your first definition becomes equal to the second when $j$ is substituded by $-j$. It is just a convention, if you consider a positive $j$ to be a up- or downscaling. 

So, when writing a paper or reading a book, it is always important to understand the definition of the transform used.