Partially, it was the "steam_appid.txt".The absence of that file causes the Steam not starting error.ORIGINAL: BertyD
Something changed, but I don't really know what. Uninstalled from Steam then reinstalled. Same behaviour when starting from Steam. But now if I run from the command.exe suddenly it's working. May have been the offline/online thing which I did after reinstalling
Maybe the on/off procedure also helped...

