If there's any network communication at all, the password needs to be hashed before being sent. Even if there's not any network communication at that level, the password should be hashed at the very first opportunity to do so, as soon as the login event is handled. Before it is sent to any sort of method that would be doing any logging.
I'm not a security expert or anything, but isn't hashing client-side completely pointless? Ideally you would send the password over SSL in plaintext and the server hashes it.
I wouldn't call myself a security expert, but I have spent several years writing applications requiring a login system professionally. You're right that in a web site, it would be the server that hashes the password, not the client. The important thing is that it would be hashed immediately, before it is sent to any other methods that would be doing logging or anything like that.
Now I don't know anything about Unity, but from what I can tell this looks like your basic Windows application. To make a parallel to a web site, I would consider the "client" side to be the front-end; where you see the window and type in your password, etc. And I would consider the "server" side to be the code behind that handles events such as clicking "login". I know it's all running on your local computer, but if you want to think of it like a secure website, then the part that handles events would be the server, and that's where it should be hashed. I assume that as part of handling the event, it calls some web service to authenticate the user. And it would just make sense to hash the password before sending it to that web service.
I admit that I have far more knowledge about web applications than Windows applications, so it's possible that I'm mistaken. And again, it all depends on where this "log" actually is.