'on my hand' describes a situation where something is on the flat of my hand, when my hand is in a situation of flatness. But if I cup my hand slightly in an attempt to hold something in, something from escaping, say, water, then it's 'in my hand'.
Same idea for chairs, on a flat or being enveloped by the chair. I'll suggest that this difference may have something to do with evoking a sensual, a more feeling nature nature;
Put lotion on my arms / Wrap her in my arms.
'He sat on the armchair' sounds to me like he sat on the arms or on the back of the chair.
'He sat in the armchair' sounds like he sat on the cushion of the chair with the arms then being around him.