Strong, good, positive, anything. I don't think it's the applicant's place to question the type of letter they will receive. It's their job to select people who they think will write them a strong letter, and then whatever letter is produced is up to the writer. Asking for a strong letter just sends the message that you're not sure if the letter they will write is going to be strong, IMO.
I agree with you a lot, but I personally have a problem with this. It's a letter of recommendation, not a letter of evaluation. If you cannot recommend someone, you shouldn't write the letter. It strikes me as disingenuous to use this position to tank people's careers, since they have no way to know what you actually wrote about them. If you don't think someone should be a doctor, you should outright tell them so and then refuse to write them a letter.
Either way, I don't think asking for a "strong" letter would have persuaded you to actually write a strong letter for these folks.
So again, I don't think asking explicitly for a strong letter is a good idea. I think applicants should just select people who they think will write a strong letter for them, and then hope for the best. That's really all we can do anyway.