Spatial Audio is one of the terms used for anything with height information: Binaural and Ambisonics (although these two don't always feature height), Dolby Atmos, Auro3D, MPEG-H or some experimental other formats.
Other terms that mean the same thing: 3D Audio, Immersive Audio, Surround+Height
Headphones are able to reproduce Spatial Audio mixes, if they've been binauralized, so the place and the system aren't too relevant (if OP already set up their playback system).
Tangential question for you then, assume the spatial audio is encoded using object based techniques (such as Dolby Atmos). Per my understanding, I need to then apply a HRTF to convert the object based audio to binaural audio for playback through headphones. So which programs do you recommended for that conversion? In Windows, Mac or Linux. Thanks