The example I have used numerous times is the cheap "Chinese" watches available at Walmart and elsewhere. The movement is made in Japan, the case is made in Brazil, the band is made in China, and it is assembled in China.
As more and more of the raw materials, components, intermediates, and finished goods move around the world, the concept of being made/manufactured in a particular nation becomes less applicable. Likewise with the movement of capital and labor
Its a world economy, or, assembly lines are out and supply lines are in.
You’re right. Vertical integration used to mean the making of a product, soup to nuts, within the same company. After a while, there was vertical integration within the same country - i.e. a product’s parts were made in one country, but by different companies. Later on, a product’s parts came to be made by different companies in different countries. The world has come a long way since the introduction of the first assembly line.