The Opera product you are thinking of is Opera Mini. Opera Mobile is a browser running mostly on your device (except for "turbo" which optimized media trough a proxy setup, but did not, afaik, execute any of the javascript).

Opera Mini can be looked at as a browser running in the cloud, sending OBML (Opera Binary Markup Language, if I remember correctly) causing the (very thin) client to draw things on the mobile screen, like text, images, etc without having to transfer, parse, execute, flow and paint every thing on the device.