I've given up on trying to use (or understand) the term "cognitive dissonance" in conversation...

I think the police officers’ use of the term is not necessarily wrong, not necessarily right. It would depend on the situation, and what the officer was actually thinking.

I think the itemized requirements of the term are something like this:

A. There are two actual thoughts.

B. The content of the first thought and the content of the second thought are mutually exclusive, i.e. if one of them is true then the other one can’t possibly be also true.

C. The same person is having both of the two thoughts at the same time, and is believing that both are true.

D. The person with the two thoughts might or might not be aware that this process is going on.

E. If that person IS aware (or becomes aware) of having two mutually-exclusive thoughts at the same time, then they will either identify it as a problem or inconsistency, and reconsider; or deny/rationalize/minimize in an attempt to continue believing that both of the thoughts are true.
I think it’s quite plausible that the training developed for the police was indeed focusing on “Examine yourself for signs of cognitive dissonance; if you find it, don’t let yourself be fooled - discover the source of the clashing thoughts, and get to the truth as soon as you can”.

And I can also imagine that some of the officers who took that training might think “I get how this works”, when they really don’t quite get it yet, and who might be using “cognitive dissonance” merely as a buzzword for “analyze your situation carefully” or something like that.

People can use words for whatever they want. But the early use of the term “cognitive dissonance” emphasized the dissonance. The painful jarringness, The thought-destroying mind-bendingness. Not the contradiction. Not the oddness.

It’s easy to believe contradictory things. Many people do. If there is no dissonance, it’s not a problem for them.

Some people have problems working things out: She didn’t love me, and she married me. Now she loves me, and she want’s to dump me. The more I believe her, the less faith I have in my own opinions. The less faith I have in mo own opinions, the more I depend on what she tells me. These contradictions are filling my head with noise and making it difficult to think.

That noise is (was) called “cognitive dissonance”, and the phrase refers not to the contradictions, but to the difficulty you have accommodating those contradictions.