In order to fix the issue, I "purged" Steam from my computer, restarted and re-installed Steam from scratch... Everything worked fine until I restarted, then boom - the icon is missing again!
Steam will launch if I run it via the respective "Terminal" command:
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steam
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Never mind.
If I manually created an item for Steam in the Menu (right-click the Menu-->Configure...-->Menu-->Open the menu editor-->category (e.g. "Games")-->New Item), it seems to "stick" between restarts (of my computer)...
The "Name" field is whatever you want it to be, the "Command" field is "steam" and if you click the icon, you can pick whatever icon you want (the actual "Steam" icon should be in there, if you want to stick with that).