what does this prompt really mean?

Your iPhone has previously been synced with a different library - which usually means one on a different computer. As the message says, you can only sync with one library at a time (and you can't select a library that's on a different PC for syncing purposes). If you completely replicate your library from one PC to another as described herethen iDevices will see these as the "same" library (i.e., two physical instances of one logical dataset) and won't erase previous device content. The key to this is that you need to copy everything in the iTunes folder (not just the media files) - the "sameness" of an iTunes library is determined by a unique identifier stored in the iTunes database file (iTunes Library.itl). See turingtest2's user tips on Make a split library portable and Backup your iTunes for Windows library with SyncToy for more info.

The current name of the computer shouldn't be significant. It could be that the library was corrupted in some fashion and a new library created with the same media. See Empty/corrupt iTunes library after upgrade/crash. It may be possible to restore a previous version of the database that is seen as the home library for the device.

tt2

See Recover your iTunes library from your iPod or iOS device just to make sure that everything you want that is on the device is secure before committing to the erase and sync.

tt2