Do I need a NAS drive for DLNA streaming my music library or is an external SSD fine?

yup you'll be fine.

Many routers these days have USB ports to support external drives, and have both Samba network drive support and DLNA server support built in. In that case you don't need a NAS, just plug in your external drive. What the router doesn't have is a lot of USB power supply, so for anything more than a thumb drive you will probably want a drive with its own power supply. Also the router USB port probably isn't very fast (even the USB 3 ports may be USB 2 speed), but for streaming music it's fine. A separate NAS is probably unnecessary for this purpose if your router is adequate.

A WD Red drive is a drive intended for use in a NAS, not a NAS itself. Traditionally WD Red hard drives are marketed as being specially designed for use as NAS drives, but in reality they are standard WD consumer drives with firmware slightly tweaked for NAS use. For example they have a longer head settling time after changing tracks because there may be vibration from other drives operating in the same NAS.