I'm sure you've already found out about this elsewhere, but I'll just post an answer here anyway for anyone who comes by: det here does not refer to the bed. It is a placeholder pronoun, used to start talking about a new topic, and for that purpose, we always use det.
Here's a link to a much longer text about this: https://www.duolingo.com/comment/9708920
I think "It is my bed" is the most general translation, though "This or That is my bed" could possibly be used in a situation where you're pointing at a particular bed and want to emphasize that "THIS/THAT ONE is mine". In that case you would put the emphasis of the sentence on "DET" in the Swedish sentence too ("DET är min säng"). Or someone who hasn't seen a bed before (I know it's quite unlikely...) asks "What is this/that?" ("Vad är det här/där?") poiting at your bed and you answer "That's my bed" ("Det (här/där) är min säng"). In this case you can omit "där/här" but as I wrote before, generally "this" = "det/den här" or "detta/denna" (not very much used in spoken Standard Swedish) and "that" = "det/den där". The emphasis is put on "här/där".
TLDR, to make it simple, both "this" and "that" could as well have been wieved as incorrect even if there are situations where they could have been used, both of them.
The difference between det här/this and det där/that is that this refers to something close to the speaker whereas that refers to something farther away. In practice of course the difference isn't always obvious but basically these concepts are used in a very similar way between Swedish and English.