All perfect praise be to Allaah, the Lord of the worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and messenger.
This is a widely circulated saying and its attribution to the Prophet, sallallaahu ‘alayhi wa sallam, is invalid and has no basis. Al-‘Ajlooni pointed out that there is no basis for the claimed virtue of having a name like Ahmad or Muhammad.
As for the last part of the question; if we presume that the Hadeeth is authentic, then it is apparent that the names Hamd or Hamad are included in this Hadeeth (had it been an authentic one).
Allaah Knows best.