The Danish sentence reads "Han lå i sengen", meaning "He lay in the bed". If you wanted to say "He lay on the bed", it would be: "Han lå på sengen". In English, to lay "in bed" means to lie under the blankets, whilst "on the bed" would be understood as just lounging on the bed.
You are correct that 'on the bed' does just mean physically 'on the bed.' But (in Australia at least), you would not really say 'lying in THE bed.' This might not be technically wrong but this would sound strange. It's much more common to omit the THE and say "lying in bed." In fact, most commonly you wouldn't even say 'lying.' You would just say "He was in bed."
I agree, as I stated in my previous comment re: " lay in bed", but the point of this exercise is to translate from Danish to English. The Danish sentence reads "Han lå i sengen", and translates to "He lay in the bed". (Sengen = the bed).
If you insist on "He lay in bed" (without the "the"), then the Danish sentence would read "Han lå i seng" (Seng = Bed").
As a further explanation, in English "to lay" is always transitive, meaning it is always done to something, whereas "to lie" is always intransitive, meaning it is always just done, and never done to anything.
to lie, present tense: "I lie on the bed"/"I am lying on the bed"
to lie, past tense: ""I lay on the bed for an hour yesterday afternoon."
to lay, present tense: "I lay my hat on the bed"/"I'm laying my hat on the bed"
to lay, past tense: "I laid my hat on the bed yesterday."
That said, even we native English speakers have a hard time with this, and the vast, vast majority of us get it wrong on a regular basis. So while "He laid on the bed" is technically wrong, very, very few people would ever even notice, and those that did notice are likely to not care.