Well! My mistake! It turns out in the top image where the font apparently works fine is because ‘World Ready Composer’ is not turned on. When I turn it on those two fonts display the same problems. Obviously the problem is with Open Type GSUB tables in the two fonts which have been poorly defined, so they work out fine in other programs like Word etc but start their shenanigans when inDesign enforces strict rules.

Delving into GSUB rules and fixing the culprit will take too long. So I just copied the glyphs onto another font that works. Looks fine except that baselines and Ascent are different. Gotta fix that now :(