"There was a police officer at school." would normally mean "When I was at school today, there was also a police officer there." Students and teachers can be "at school". It refers to both the time and the place. If I'm talking about a school that I am not connected with, I would have to say "There was a police officer at the school today." If I'm talking about the school I attended, I could use either phrase but they would have slightly different meanings. "the" makes it more objective and less personal.

Correct! You can either say "at school" or "at the school." Either is correct. And it doesn't matter who you are talking about. You could even say, "There was a cat at school" or "There was a cat at the school" and either would be correct as well.

Technically, you are correct. "There was a police officer at school today." vs "There was a police officer at the school today." These could mean they joined your class to learn. But, overwhelmingly this is unlikely, so we would have to assume that it means visiting for a policeman. . I saw the janitor today at school. He normally does not appear in our building during school hours. There was a fire at school today. :) There was a ruckus at school today.