For reference these are my tapping heads:

The Tapmatic (bought new) and the Edelmatic (Ebay) have built in reverse so the machine spindle doesn't need to be reversed. On the manual mill the quill is used to feed the tap, when the stop is reached the forward clutch disengages and the tap stops. A sharp pull up on the quill engages the reverse clutch at twice the spindle speed. On both heads the large knurled ring at the top is for adjusting the torque at which the down feed clutch will slip.

The head on the right is a tension/compression one (bought from Maritool in the US) for use on the CNC mill. It requires the machine spindle to be reversed to back out the tap. The feedrates can be finessed, a bit over nominal on the way in and a bit less than nominal on the way out. That way the tap is pushed into the work slightly by the spring in the head and on reverse the tap will cleanly pop out from the hole so the beginning of the thread starts cleanly and isn't damaged on exit of the tap.

Andrew