A player in our campaign asked if they could change a ship from 1 variant to the other variant and pay the point difference in the base cost of the variants.

I say no because there isn't a mechanic present which would allow such an extensive refit of a ship. The closest thing I can find is on pg. 8 of the current FAQ where it states that players can remove a ship/squadron from play but they still count as destroyed. They won't recoup any of that ships costs either. So this player will need to remove the variant from his list in during the refit and expanding fleets phase then he can use resource points to buy a new ship but the variant he wants?

Am I correct in my understanding?

Not by the default rules (the upgrade part). You cannot change the ships type (upgrade a ISD I to ISD II by spending 10 points).

He can remove the ISD I and buy an ISD II for 120 points. If he really want. But this would be a, lets say it nice, stupid idea.
It might be an option later in the campaign, when a player has enough points and want to change the fleet. And it might be a good idea for anything cheaper.

You and Tokra are correct.

However, many of us play CC with a list of house rules. In the campaign I'm currently part of we allow a ship to change varient, but at the cost of half of the new varient's point cost (rounded up). So, for example, to change from an ISD II to an ISD I would cost 55 points. Or vice versa would cost 60.