No, M-phase is not of three types. M-phase or Mitotic phase is a phase in which karyokinesis is followed by cytokinesis. Futher karyokinesis comprised of four different phase i.e. prophase, metaphase, anaphase and telophase.
Meiosis is a different type of cell division mainly occurs in germ cells to produce gametes.
Amitosis is the type of cell division which involves splitting of elongated nucleus and followed by cytoplasmic division.