As a graduate student, is it better to use a Gmail address or name@mywebsite.com?
I don't know how people view things in the humanities, but in mathematics having your own domain named after you sounds a little presumptuous, at least to me.  Some people do, and I find this normal for people like  writers or some programmers, but not most types of academics.  In addition, sometimes these go defunct because people stop renewing them, so something like gmail is a more "stable" solution.  If you dislike gmail and don't want to use your institutional address, check out some other email providers (yahoo, zoho, mail.com).

Note: to me, instituional addresses still seem more "professional" than something like gmail, but sufficiently many people use gmail for professional reasons that it doesn't look strange to use it.  In fact, something like gmail is often a better choice for young people who haven't settled down and may change their institutional address several times.