A Livechat inquiry is an internal data structure used to control the Omnichannel queue. The conversation state - ready, queued, or taken - is a state of the conversation in the queue, and data is stored in the inquiry collection. It won’t be useful for the bot integration because the Omnichannel routing system will handle all stuff.
Omnichannel is a product name. In older documentation, it is referred to as Livechat. In recent years, the former Livechat feature has been expanded to include Whatsapp and Facebook Messenger. In the near future, the Omnichannel product may include interoperation with other technologies such as Viber, Wechat, Blackberry, Apple Business Message, and RCS.