An answerer on this question stated "remember to never use snakes on toilets." The internet, however, contains numerous tutorials on how to snake a toilet.
Can and should toilets be snaked?
There are snakes designed for clearing clogs in a toilet. They typically have a 3-foot tube with a rubber piece on the end, and the spring part goes through the tube and the rubber piece. The rubber protects the toilet from being chipped or scratched by the snake.
The type of snake talked about in your linked question are designed for use in cleanouts where the is no p-trap. Using one of these snakes in a toilet or a p-trap could scratch, crack, chip or break the toilet or p-trap. I have seen a plumber pulling on a fat snake that was down a toilet actually shatter the bottom of the toilet.