The exclusion of any mention of magical weapons isn't accidental, Barbarians are resistant to bludgeoning, piercing, and slashing damage whether it's magical or otherwise as it's key to their ability to have staying power on the front line. By comparison, the Heavy Armor Mastery feat makes the clear distinction for magic weapons, therefore, it's not unreasonable that the design intent is on purpose.

This is important because otherwise the Barbarian would experience a huge drop off in survivability as the game progressed passed a certain tier when a lot of enemies' weapons are being described as inherently magical.

Going off the phrasing of the effect

When the Lich takes damage

Not "When the Lich is hit with an attack/spell/etc," when they take damage.

In this case, a Lich's resistances and immunities are applied before you determine whether or not Tether fires off. Because until you apply those, the Lich hasn't taken any damage yet.

So, since when you hit a Lich with a nonmagical sword they take Zero Damage, then even if you still ruled that Tether activated, then the Wizard would have to save against taking half of zero damage...so, no damage at all.

The specific wording you are referring to of

Resistance and then vulnerability are applied after all other modifiers to damage

Are all still applied before a creature actually takes any damage. It's not like a Fire-immune creature gets hit with a fireball, takes the fire damage, and them immediately heals that damage. They simply don't take any damage at all.