Finally getting caught up.
Hoist is a wheel lift wrecker (my dad had one of these, also why it disappoints that hoist has never been a Wrecker), it lowers down bars that stick under the car and the goal is to go around the wheels and lift up from the wheels then pull the car. I have never understood what the wings were for other than a light bar. The best I can come up with is a mix of the light bar and a representation of the pulley system.
You're not going to see the tow mater style hook on any recent tow truck because that style opens you to liability for damage to the car. Either you're going to pull off the bumper or bend the frame. You use it to pull cars out when you're not in reach for the bars (likely already totaled). The style my dad had did both, it had arm bars that swung down, but also had various attachments like a hook to pull out cars.
https://www.trustworthytowing.com/towin ... cks-types/