Wizards are indeed one of the common characters of fantasy and usage of the term has varied over time and between authors. I remember in the fantasy books in my father's collection there could be good or bad wizards, and they were only male. Women could do magic, but they were called witches or sorceresses, both of which terms usually have a negative connotation. The Oz books did have Glinda the Good Witch of the South, who was also called a sorceress, but she was one of the exceptions of that time period. J.K. Rowling had her Hogwarts School of Witchcraft and Wizardry. Andre Norton had her Witch World series, and other books in which female magic users were also called Wise Women. When I started writing my fantasy books, though, I noticed very few male magic users were ever called "Wise Men". "Wizard" was the term of preference and that was the term I decided to use for my female magic users. Other authors decided the same. Diane Duane in her Young Wizards series had both male and female wizards, as did Diana Wynne Jones in many of her books, while Tamora Pierce in her Circle series had both female and male mages.
Wizards don't always have to be human, either. I have elf, gryphon, and even horse wizards. Mercedes Lackey has gryphon wizards while Diana Wynne Jones has gryphons, centaurs and even dwarves becoming wizards. Diane Duane's wizards include not only alien centipedes and trees but also a planet.
Returning to the "Need a Wizard?" panel, it wasn't until I read The Lord of the Rings series that I encountered the idea that wizards could specialize. Radagast focused on animals, Saruman was the craftsman and metal-worker and Gandalf ended up as the warrior. Jim Butcher's Harry Dresden was one other wizard mentioned at the panel as another warrior. In Diane Duane's Young Wizards series, Kit's magic works well with machinery, while another specializes in stellar engineering. Tamora Pierce's Circle series has mages who can control storms, plants, metal and even thread.
What types of wizards do you enjoy reading about or writing about? What tropes have been overused?