A restaurant offers "Takeout or Pickup" and it appears the difference is that takeout are orders placed onsite to be consumed offsite, and pickup are orders placed offsite that are retrieved from restaurant to be consumed offsite; delivery being orders that are both orders and consumed offsite? Is this correct, what is the history of the use of these terms?
The words are used in different contexts.
When you call in an order by phone, it's clear that the food will be consumed offsite. However, the restaurant needs to know whether you will pick it up yourself (pickup) or you want them to deliver it to you (delivery).
When you place the order at the counter, delivery is not a possibility. In that situation, they need to know whether they should serve it on a plate so you can eat it in the restaurant (eat-in) or package it so you can take it out of the restaurant (takeout).
When explained this way, the origins of the terms become fairly obvious. They're just abbreviated forms of the actions that will be formed.