These terms are usually referring to the same Purim custom: Machatzit hashekel, Machatzis hashekel, Machzis hashekel, Mahatzit Hashekel, and Maasit hashekel. The standard expression in English is Zecher Machatzit HaShekel.
It is a pre-Purim donation custom done as a remembrance of the half-shekel contribution once given in the time of the Beit Hamikdash.