No, 10K is not short. 10 K is a race, I do a couple of 10K races every year although my primary focus is marathon. I don't like half marathons for multiple reasons: it's an awkward kind of effort when you're in for 7X minutes - just a little too long to sustain at LT pace. Most of the races are not serious, every village runs a half marathon where half of the field hardly even train and RDs often do half-assed mistakes like mis-measured courses. It's not even an event in the Olympics.
10K s are almost always real races with serious runners and hobbyjoggers unless they are 'companion events' to a city marathon.
Now 5Ks are short and fast. I just can't run that fast when I'm training for a marathon (which is, almost always). I do 1-2 minute intervals at 5K pace as my speedwork and such sessions have me trashed.