How to use iTunes library on a different mac stored on external hard drive - on wired network

Why not just unplug the external hard drive and move it? And I don't believe so. You can use Home Sharing to access the library and iTunes Wi-Fi sync by following the directions in the link.

Share your iTunes library over your home network - Apple Support

Sync your iPhone, iPad, and iPod touch with iTunes - Apple Support

Best regards,

Austin

I use Apple's iTunes Match service ($25 per year) to do what you describe, with additional benefits. My primary Mac has all of my song files stored locally (on an external drive), and that Mac was used as the starting point for my iCloud Music Library, which iTunes Match provides. My other Macs access that common iCloud Music Library and stream songs; they do not have locally stored song files (although they could). I can update my iCloud Music Library from any Mac, and the other Macs immediately see the change. One of them is quite old (a PowerPC G5) running iTunes 10.6.3 and Leopard (10.5.8), and iTunes Match is supported. I can also access the same iCloud Music Library from my iPhone and iPod touch, to either stream or download songs (and make updates).

I recommend keeping a local copy of ALL song files on one of the Macs, instead of only having your song files in iCloud Music Library. And also back up that local copy of song files using Time Machine, along with your OS X system and other user data. Here's Apple's info on iTunes Match

http://www.apple.com/itunes/itunes-match/

You can use iTunes Home Sharing, but the "other" Mac cannot make updates to the shared library. You can probably set up File Sharing between Macs, so that the "other" Mac accesses the same iTunes Media folder (on the external drive connected to the "primary" Mac), but the two Macs have their own separate iTunes libraries; it may be a pain to keep them "in sync" manually (plus the sharing Mac needs to be ON all the time). With iTunes Match, all associated computer and devices share the same iCloud Music Library.

What exactly do you mean when you say library? A library is an entire iTunes folder with media files but also all the support files and folders. You can copy everything except the media to the new Mac, then start iTunes while holding down the option/alt key and when prompted guide it to the iTunes Library.itl file you just moved. It should "point" back to the media on the original drive as long as it mounts on the new computer as a drive. If there are problems and you get broken tracks ( ! ) you can find one track when asked, then let iTunes try to find others based on the one you find (will take some time).