Reporting back:

I called the operating carrier, and my eTicket is still showing the unchanged departure details on the reservation agent's end.

Upon learning this, I did some more poking around. When I view the reservation online, the departure details look correct, on both the operating carrier website as well as Avianca's website. However, when I request an emailed eTicket from the operating carrier, it shows the wrong departure details again.

The rebooking tool did something, but it did indeed seem to leave part of the job undone.

Out of sheer curiosity, and because this flight isn't for several months, I'll email LifeMiles and see if they can fix this over email.