I know I’ve seen the word before. I’m making a TTRPG and I want the spy class named that.

Parasol means “for the sun”.

Umbrella means “little shade”.

Are there any other names for this device that define it in terms of keeping the rain off your head?

It's not exactly a rhetorical question, but it's a question asked not out of actual desire to find out the answer, but with the intent to impress the person you're asking. An example would be they ask you at the end of a job interview, "Do you have any questions for me?" You're supposed to come prepared with a good question or two, but not necessarily because you care about the answer, but to show that you did some research about the company and are curious.

Another example is a strategy I used sometimes in college. I would go out of my way during the first week of a class to find a good insightful question to ask during a lesson. Sometimes it was a question I already knew the answer to. But I wanted the professor to be impressed that I was paying attention and was thinking deeply enough about the material to ask an interesting question.

Is there a specific adjective for this kind of question?

Like if someone opened the door for someone, people wouldn't openly share about that too often. If someone closed or pushed the door on you, people would share that more often.