Apple - How do I enable "Network Trash" on a file server?

The Synology NAS supports a Network Recycle bin, but only for SMB volumes. Therefore if you want to make use of it from your Mac, you need to access the share via SMB and not AFP. This can be enabled from the Control Panel under Win/Mac/NFS.

The Synology by itself fully supports Time Machine, and can act as a Time Capsule. Most of the documentation on the Internet mention the Network Trash and Time Capsule option in the same place, but as an Windows/Mac comparison type description. The manual for DSM does not mention Network Trash for Mac either, only for Samba.

I am not entirely sure where you read that the Synology supports this feature for OSX, since I couldn't find any reference to it in the way you described.

OSX, unlike the previous OS 9, does not support a Network Trash of any kind, and I am not aware of any plans to bring this feature to OSX. Therefore it also can't be activated from the OS layer.

I suspect the naming was changed from Network Trash to CIFS Recycle Bin in the new DSM by Synology because of the confusion it has caused.