I have been wondering about this as well. Betjent to me sounds more like servant. If I wanted to say military officer, how would I say that?
You are right. "Betjent" is a variation of the german "bedienen". An officer in the army is "en officer i hæren"
Police officer usually. Could also be certain other functions like "ranger" - "parkbetjent".
To be absolutely sure you need to specify "politibetjent".
Military officers are called "officer(er)" in Danish too. They are not betjente.
In my Danish classes, we learned that professions are always without articles. Is it that it is an old-fashioned rule going out of use?
If you say "jeg/du/han/hun er betjent" you do not need the article and it sounds better without it.
If you say "jeg ser en betjent" (I see an officer") you cannot drop the article.
If in doubt I would suggest using the article. Having one that was not needed is better than forgetting one.