Unfortunately the mechanic didn't express the issue very well to you, or used vocabulary that you haven't come across before, but the best person to ask for clarification is that mechanic. He is the only one of us who has been hands on with the bike.
I would expect normal headset adjustment to be part of a tune up, and the fact you say he tightened something, would be just that, tightening the preload bolt with the stem loosened. I suspect that because you only took it in for a tune up, he didn't take the assembly apart to do further diagnosis. He certainly could have done, then charged you for the time it took to fix it and the parts required; your bill would have unexpectedly doubled and he would have lost time for other work when bike shops are under unprecedented demand. Can't please all the people all the time.
It's unlikely that there's damage to the bike that can't be rectified quite easily. There are self-contained bearing units top and bottom of that head tube which can wear but also potential for dirt, grime, corrosion etc to prevent the precise fine tuning of the preload. It might just need taking apart, cleaning and greasing and putting back together, or a new bearing assembly dropping in.
Something you can try is to try and twist the three carbon spacers just under the stem itself. If they rotate then there is probably too little preload, but if they don't rotate then they are clamped tight, and the play you feel in the headset is more mysterious