If using an external HDD formatted NTFS (but not FAT32) on different computers, the PC it was originally used on has set all the security identifiers on all the files. When you move the HDD to another machine, it cannot understand which users have which access, because it does not recognize the security identifiers. Hence it asks for Admin rights to be able to write to the drive.
Usually you can still read the HDD on any other machine. This is why FAT32 is often used for external HDD, it does not carry those security identifiers, making the HDD usable across multiple PCs.
Another factor may be that the Library PCs have registry entries set to disable external USB devices. This is commonly done to prevent loading of viruses, etc. on the Library machines. Thus your external HDD may not work at all on those machines without admin rights (or they tightened USB security recently - ask them...)
So, without knowing the entire usage (and formatting) history, hard to say why it worked before, but not now. I suspect one of the machines may have set security on some files/directories recently, or the Library machine disallows USB device usage.