Definition of weight $k$ Laplacian

Question

I just found that there are two different definitions of the weight $k$ Laplacian on the complex upper half plane. In Bump's book, he defines $\Delta_{k}$ as $$ \Delta_{k} = -y^{2} \left( \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}\right) + iky\frac{\partial}{\partial x} $$ but Wikipedia define it as $$ \Delta_{k} = -y^{2} \left( \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}\right) + iky\left(\frac{\partial}{\partial x} +i \frac{\partial}{\partial y}\right) $$ and I found some papers that use the first one, but also find paperst that use the second one, so I'm very confused now. Which one is the right one? I think it may depend on the context, but I'm not sure.