Make sure that you have
ProjectionType="cylindrical" rather than
ProjectionType="equirectangular" in your EXIF. You must also specify the HFoV and VFoV (horizontal and vertical fields of view) for the image. The documentation you linked to has the following example exiftool command to define a cylindrical panorama:
# 120º h_fov, 86º v_fov
exiftool -FullPanoWidthPixels=18000 -FullPanoHeightPixels=9000 -CroppedAreaLeftPixels=6000 -CroppedAreaTopPixels=2350 -CroppedAreaImageWidthPixels=6000 -CroppedAreaImageHeightPixels=4300 -ProjectionType=cylindrical file.jpg
A 360ºx180º equirectangular panorama is a full 360º spherical view that includes the top and bottom of the sphere, that's why it's interpreted as such by most viewers. There is no such thing as a 360ºx360º pano.
The panorama format that's limited to side-to-side pans are cylindrical panoramas, which typically don't have a 2:1 aspect ratio, but are usually quite a bit wider and shorter, say, 4:1 or 6:1 aspect ratio. The Mark Zuckerberg Facebook example you link to is a cylindrical pano.
A cylindrical pano usually looks like a long thin horizontal strip when viewed all at once. But more importantly, it is completely missing the zenith/nadir information, so you cannot tilt up to view the ceiling or down to view the floor.
Not all pano viewers can handle both equirectangulars and cylindricals. But if FaceBook's can, it's also probable that simply making sure your panorama does not have a 2:1 aspect ratio is all that's required.