Another change involved when the Kaishakunin (the second) was to deliver the daki-kubi (coup de grâce).
Initially (back when being Samurai really counted for something) the Kaishakunin was to wait to strike until the condemned had completed his cuts (the number and location of which also changed over the centuries) and returned the tantō to the tray it had been offered on. In the latter Samurai era the ritual was watered down so much that all the condemned need do was to stretch out his hand as if reaching for the tantō and the Kaishakunin would strike.
Being Kaishakunin was both a great honor and a great responsibility because the condemned's obligation (to his lord, etc) was considered fulfilled once he had completed the ritual cuts. The Kaishakunin's role was to end his suffering once this ritual duty was dispatched.
And it took a swordsman of considerable skill because his duty was not to decapitate the condemned but only to sever the spinal cord but leave the head still somewhat connected to the neck. Depending on the source, this was done because it would have been in bad taste for the condemned's severed head to roll out amongst the guests present, or because if it fell off and rolled around, the head might become soiled. Since the condemned was restored to an honorable status at the moment of death, it would have been a dishonor to allow his remains to become soiled.
Done to best effect, the head would fall into the lap of the condemned, who would be sitting cross-legged in the lotus position. That way it would appear he was holding his own head.
Hadn’t read or heard of the lap bit, but had read that the best kaishakunin left the head on with just a fold of skin in the front still attached to keep it from “rolling”.