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I found relevant discussions on KDE phabricator Add support for icon-only tasks (what browsers call pinned tabs) and patched KDE (plasma-desktop and plasma-workspace packages on Arch Linux) using relevant diffs from there.
This adds an option to show icon only for specific apps (under "More Actions" → "Show the application icon only").
Now Task Manager looks like this:
Also make sure to uncheck "Keep launchers separate" in Task Manager settings, otherwise icon-only entries will be mixed with normal entries
Welcome to Super User! Your question is a bit unclear to me. Which app did you start that made the labels to appear? Do you want that only the icons of the pinned apps are shown instead of the titles? – Aulis Ronkainen – 2018-12-18T19:54:10.473
Any app pinned in KDE task manager widget appears as icon on taskbar but when launched, this app window will appear in task manager as a regular entry (which has icon and window title). I want to see only icons of the pinned apps even when they are running (aka icon-only task manager for all pinned apps + regular task manager for all other windows). – krelly – 2018-12-18T20:21:50.273