What are the rules for upgrading existing magic item?

This question comes from the comments on my previous question.

In short, if I have for example a Belt of Incredible Dexterity +2 (4,000 gp), and I want a Belt of Incredible Dexterity +4 (16,000 gp), can I simply go to a wizard and pay the difference and wait for the necessary days to have the item upgraded?

In this case, since the cost goes from 4k gp to 16k gp, I’d have to pay the wizard 12k gp and wait 12 days, while s/he would have to spend 6k gp to perform the task and keep the other 6k gp as profit.

However, the only example I found in the rules, is of a wizard upgrading his/her own item:

Example: Patrick’s wizard decides to use his downtime to increase the armor bonus on his bracers of armor +1 to +3. The price difference between the two types of bracers is 8,000 gp, so Patrick’s wizard must spend 8 days and 4,000 gp (half the 8,000 gp price difference) upgrading his bracers’ magic. If he has fewer than 8 days before the next adventure, he’ll need to finish his crafting while traveling or use accelerated crafting in town to speed up the process.