I don't know why the shop did it wrong, but we can walk you through doing it right. You have a flat spanner that fits on the adjustable cup, right? It should look like this.



You also need that hooked spanner, so yes, go ahead and buy it. You might be able to get by using a big channel lock instead. You'll mar that notched nut, but maybe you don't care. I would do it with my bike, but some won't do it with theirs.

The notched thing is not a washer; it's actually a nut. It's somewhat redundant, as most headsets just use the top nut as a locknut.

Put the bike on the floor. You don't do this adjustment with the bike in a stand. Face the bike from the front, and put the front wheel between your legs. Hold the headset wrench with your left hand and the hook wrench with your right hand. Adjust the cup approximately and tighten the notched nut onto it. Do this by holding the wheel with your legs, hold the cup still with the wrench in your left hand, then turning the notched nut with your right hand. Once the notched nut is tight against the cup, use your right hand to apply the front brake. Rock the bike back and forth to see if there is play. If there is, you need to repeat and tighten the cup. If there is no play, lift the front of the bike and make sure the cup is not too tight. You want the front end to spin easily. If it is too tight, redo, loosening the cup. It takes some trial and error.

Once it's adjusted with no play but also no binding, hold the wheel with your knees, hold the cup with the flat wrench, and tighten the locknut with a big wrench.