Do you get Hex Warrior’s Cha bonus with a two-handed Pact Weapon?

Hex Warrior says:

you can touch one weapon with which you are proficient and that lacks the two-handed quality. When you attack with that weapon you can use your Charisma for attack and damage rolls.

later the ability says:

This benefit extends to every weapon conjured with the Pact of Blades feature, no matter the type.

(emphasis mine)

Does this allow me to use my charisma with 2 handed weapons(provided it is my Pact Weapon and my Hex Warrior Blade)?

This question is not the same as Hex Blade Warrior & Pact of Blade. That one is asking how many weapons, I’m asking if type means 2 handed or something else(Slashing or Ranged for examples).