How to write to NTFS drive

I am surprised that your MacBook Pro allows you to write into that external drive without any third-party app. That makes me think your Seagate drive came with a driver that allows you to write in NTFS devices. Do you still have that Seagate software? If so, try installing it on the Mac mini.

You need 3rd party software in order for Macs to be able to write to NTFS drives. That's you're only option. Install the app of your choice on your Mac.

Remember, you will not be able to store or use an iPhoto Library on that drive. The file system does not support the complex file system of database files and other files that make up an iPhoto Library. Doing so will corrupt the library. The library must be located on an OS X Extended (journaled) partition.

One option would be to partition the EHD into 2 partitions, one OS X Extended (journaled) and the other FAT 32. The Mac would be able to read and write to both partitions while a PC would only have access to the FAT 32 partition.

Don't use NTFS, use exFAT... both OSes can natively read/write to that format and it's suitable for larger disk drives.

Your best solution would be to reformat the drive in exFAT (of course you will need to move the content currently on the drive to either your Mac or PC before doing so).

I haven't copied any photos to the drive yet so if I have to reformat it or partition it, that'd be no big deal.

Remember the limitation is for the iPhoto Library package. You can copy and store image files, i.e. .jpeg, etc., to the drive OK.

My problem with third party add-ons is that they can break whenever Apple does an update to the software. It is always better to use a format that the OS can read/write to natively than rely on a third party add-on. You gain NOTHING from using NTFS from a Mac point of view... that format was written specifically for Windows and only Windows can take full advantage of it. Just like only a Mac can take full advantage of MacOS Extended.