RAW/RAI:
1. Sheathing a weapon is an object interaction.
2. Drawing a weapon is an object interaction.
3. Dropping a weapon is not an object interaction.
4. Picking up a weapon is an object interaction.
I agree with all of this. Unfortunately the result of such is that if someone wants to cast, needs the free hand to do it, yet wants to remain armed (e.g., to enable opportunity attacks), that results in the following actions:
1. Free drop the weapon
2. Standard action cast the spell with the free hand
3. Use the one interaction to retrieve and re-equip the dropped weapon.
Because I feel that is an unrealistic result (the constant dropping of the weapon for free to work the system of actions), that is why I instead let PCs slip the sword into the shield hand briefly or tuck it under an arm, etc. and then re-equip it after, as the interaction, since that seems more reasonable that constantly dropping and picking up a weapon. It may not strictly be RAW, but it is more RAR (rules as reality).
I do like the even looser approach of if the caster is not bound, then they can do somatic actions without worrying about all this juggling, but (a) that even further demeans the War Caster feat (at least under my approach, that bullet for War Caster is still useful for casting reactions while holding 2 items), and (b) loosening about somatic actions just affects spells, and I'm concerned about broader actions than that, including using thieves' tools reasonably for the bonus action lock-picking, etc., so a rule just loosening somatic actions doesn't really accomplish what all I need to rules to actually work.