This idea is tied to wealth, if you live in Asia and you are poor, work the fields, you will become tan. Though if you are wealthy you will be able to work inside and/or not work at all.

In the West, post industrial age, being tan is sought after because you are not working, you are on vacation in the tropical islands or relaxing in paradise !

To add onto this question, was the preference for light skin in India at all influenced by the Indo-European, Greek, Mughal, and British invasions by light-skinned peoples?

In western culture it goes quite far back, at least until the early Middle Ages. Lighter skin meant that a person didn't have to work outside and get tanned by the hot sun like a peasant did. So white skin became equated with the upper class and tanned skin with the lower class. Since the upper class traditionally set fashions, the desire for white skin became associated with the standard of beauty.

As old as light-skinned people who have had contact with dark-skinned people, I'd have to assume.