I know that I've read about a number of biases by now, but they don't come to mind very easily. If I wish to become wary enough to spot cognitive biases in my own thought, then I might appreciate being able to quickly summon many examples of cognitive biases to mind.

You probably don't need to be able to make a full list of cognitive biases off the top of your head; rather, you need to have relevant memory about biases triggered whenever you encounter a situation where you are prone to them. As for training this, you can, say, open a list of biases at the end of the day, and figure out which biases you were affected by during the day, and try to remember those.

Of course, if you often want to tell others about cognitive biases, memorize a few of them with some good examples.