I've been having some issues with tight headsets lately and am seeking some guidance.

My technique for checking headsets is to brace the bike against the floor (with my foot on a chainstay) and moderately jerk the front wheel and the top of the stem in contrary directions with my hands. I believe this allows me to exert a good amount of force on the headset and detect even small knocks/looseness. The problem is that I can't seem to eliminate the tiny knock I can almost always detect without tightening the headset down pretty tight -- tight enough that I can feel the bearings in the headset and the bars don't turn 100% freely.

So my questions are: Am I applying too much force when checking my headsets? Am I correct that there should be absolutely zero play in a proper headset adjustment? I'm assuming that it's better to err on the too tight side here, correct? Thanks.