Are you allowed to play a monster in face up defense position?
Is that correct?


Yes.

When you normal summon or set you can only do it in face up attack position, or face down defense position, respectively.

There is no thing as "face down attack position". There was before, on the Darkness Approaches card, but it had an errata that removed that effect.

You could however special summon a monster on face up defense position if it is possible. One example may be any Synchro, XYZ, or Fusion Summon you carry out, where you can select how to place that card (unless otherwise restricted).