As you go on to say, you CAN tell the difference between overworked and lazy. But when you leave a small tip for bad service, the server probably just writes you off as a low tipper and doesn't consider that the service was poor.
When we go out and the service is awful, I still leave the 20%, but I don't leave it with the bill or on the table. I find the manager of the establishment and leave the tip, along with my complaints.
And I always say "when he/she tells you later that we didn't leave a tip, you can tell them why". I have yet to get bad service a second time.
Anyone who can count the number of waitstaff and the number of customers without removing his shoes can make that determination. My baseline is based on a reasonable expectation of what can be expected given the circumstances.
But when you leave a small tip for bad service, the server probably just writes you off as a low tipper and doesn't consider that the service was poor.
In almost every case that I've tipped below 15-20% (or not at all, in extreme cases), the cause has been abominably slow service (not caused by overcrowding, for which I make allowances). If the staffer can't make such an obvious connection, then spelling it out isn't likely to help.
The one time in my middle-aged life that the service was truly bad, a nickel did my talking for me.
That way, (s)he can't mistake me for a low tipper OR (in the case of no tip at all) a forgetful or penny-pinching boob.