I see from also searching online as you did that most people seem to think of character sketches as just filling out forms or writing a straight description. But I remember writing "character sketches" in college and we basically wrote "mini stories" as you've said. I think I wrote a sketch describing a man's character using his hat as an analogy for his personality and life.
As I understand it from my college classes (throwing aside what Google says), a character sketch is like an artist's sketch. It's a loose rendering of a character without the formal trappings of a book or story. Think of a character sketch as proof of your character's personality. First, create the character, decide what they look like, what their personality is like, what their wants and desires are. When you've done that I can see this going one of two ways:
Describe them as you would if introducing them in a book. Don't just give bare facts, of course. Show, as you say, don't tell. But you don't necessarily have to have the character doing anything. Show their personality through their possessions or their room or the way they move or sit or some other way.
Do plop them into a mini-story. Put them into a situation that will show that character's personality without telling about it directly.
For instance, in Harry Potter, Rowling could have just written "The Dudleys were cruel, vindictive, heedless, stupid money-grubbers," but instead she showed us these traits by their actions and responses in the first couple chapters (of pretty much all seven books). These chapters give us a character sketch of the Dudleys. But the first chapters aren't actually a story in themselves.
Whichever way you do it, if I were your professor (which I'm not, so take this with a grain of salt), I would want to read your character sketch and be able to know the sorts of things about the character that I would know from people-watching a real life human.
I write a lot of descriptions and character sketches as practice. You might try reading this one about Diana and the Frog for an example of number 2 and see what you think Diana is like as a person. Keep in mind that I didn't bother to include any physical description of her, which you'd probably need.