There are TWO "right hand rules" (well, scores, if not hundreds, but the direction of magnetic force induced on a positive charge isn't relevant to this discussion).
One asserts that vertices be ordered in rings so that, if you walked the perimeter, with one hand within the figure, and one hand outside, that the right hand be inside: the exterior ring is clockwise. Shapefiles use this right hand rule (file and enterprise geodatabase use the left hand form of this rule natively, though the SDE API libraries allow for extraction in either order, and the File Geodatabase API uses the shapefile geometry specification for coordinate encoding).
The other rule asserts that if you place a balled fist on the plane of the geometry, and extend the thumb upward, away from the face, then the exterior rings will follow in the rotation of the unclasped fingers (counter-clockwise for: the exterior ring) is counter-clockwise. The KML specification uses this right hand rule (which would be considered left-hand if walking the perimeter).
It appears that the first reference to "right hand rule" by the GeoJSON spec refers to the thumb/finger-curl rule, and the erratum refers to the body-on-boundary method. I was only taught the body-on-boundary form, but I've met engineers who were only taught the hand-curl form.
The nice part of the first example is that it applies to interior rings ("holes") as well, without the need to assert that interior rings are wound in the opposite direction of exterior rings. For what it's worth, the GML specification is explicit that the winding order isn't required to be in either handedness, but does insist that interior be opposite of exterior, and that interior rings immediately follow their exterior ring.
The specification you cited states:
Note: the [GJ2008] specification did not discuss linear ring winding order. For backwards compatibility, parsers SHOULD NOT reject Polygons that do not follow the right-hand rule.
so it's probably not worth getting worked up over. If you use the Shoelace Formula (or the Trapezoid Rule) on the vertices of a ring, the area will be of opposite sign for exterior and interior rings. So long as you place the interior ring vertices following the exterior ring to which they define an exclusion, most parsers should be able to handle your input.
One place where difficulty may occur is if the ring touches itself at a point. In some libraries, the enscribed shape may be encoded as a hole, but Esri's libraries consider this an "inversion" and handle it as described in this question.
