Apple - How to disable Notification for Terminal app?

Whilst it's not ideal, if it's that bad you could disable the app bounce notification completely for all apps:

defaults write com.apple.dock no-bouncing -bool TRUE && killall -HUP Dock

^{To revert, replace TRUE with FALSE.}

You can provide feedback to Apple regarding this at the following page:

• http://www.apple.com/feedback/macosx.html

I am running the Terminal version 2.9.1 and macOS Mojave 10.14.3 and I could just do this in the preference setting of the Terminal App:

Terminal > Preferences > Profiles tab > Advanced tab > Uncheck everything under the Bell section

Unchecking 'Bounce app icon when in background' option would prevent jumping of the terminal icon.

Running your commands in gnu screen can solve your problem.

Steps to replicate bad behavior:

1. $ sleep 5; tout bel
2. Command-Tab away from Terminal
3. Wait

Observed behavior: Terminal icon gets badge and bounces

Desired behavior: No badge or bounce

Solution:

1. $ screen
2. Control+A, c # create a new screen window
3. $ sleep 5; tput bel
4. Control+A, Control+A # switch to screen 0
5. Command-Tab away from Terminal

New observed behavior: No badge or bounce

Note that you'll still get badge + bounce if the screen window is the active one. I'm sure you can refine this further.