How to stop iPhone from "erasing its contents" when I want to "manually manage" it?
The solution is a two part solution.

If you routinely want to sync one device with two libraries, the easiest manner is to disable all automatic syncing on the first iTunes library before you bring the iOS device to the second computer.

You can always wait for the "erase" warning and cancel the sync. (which is the point you have in your screenshot). At that point, go into the current device settings and turn off automatic sync for all of the categories that cause the warning dialog to appear. Keep in mind, that music, ringtones, and other settings can all get the warning to pop up, so your work may not be done by simply un-checking music. The settings on the computer you get the warning are in conflict with the settings from the previous computer so you have to reconcile that before the manual sync can start.

Once you have figured out all the "silos" that matter to you, you can easily take one device to multiple computers for backup but only certain computers for syncing. Using one computer to sync the data (contacts and calendar) and another for the media is a use case that Apple fully supports in iTunes. Once you set things up, you should not expect to have issues with a one device and many computers situation.