Ticks love this cooler weather. Sometimes when removing ticks their heads can pull off and remain in the skin. Ticks prefer not to have their heads torn off and they most often will try to release when pressure is applied. What they seem to have trouble releasing is heir mouth parts. When you remove a tick and can still see something black in the exit place, it is much more often mouth parts than a head. If it is mouth parts or a head, they cannot live without the rest of the body for very long, The mouth parts are soon dead and he body will just reject them, almost no extra reaction is seen. If, indeed, the head is left, it will soon die. Because there is more tick issue left in the dog there is usually more of a reaction--a larger bump, maybe some redness. In either case there is no emergency. You can clean the area with hydrogen peroxide, maybe even use a warm compress. It should be fine, but there may be a bump here for a week or more.
There is no need to spend time trying to remove it, likely you are only creating more trauma. Let it be for now.
Hope this helps, thanks for your question.