Holy Lance
The name of the soldier who pierced Christ's side with a "lonchē" is not given in the Gospel of John, but in the oldest known references to the legend, the apocryphal Gospel of Nicodemus appended to late manuscripts of the 4th century "Acts of Pilate", the soldier is identified as a centurion and called Longinus (making the spear's Latin name "Lancea Longini").
Makhaira
The Koine of the New Testament uses the word "makhaira" to refer to a sword generically, not making any particular distinction between native blades and the gladius of the Roman soldier. This ambiguity appears to have contributed to the apocryphal "malchus", a supposedly short curved sword used by Peter to cut off the ear of a slave named Malchus during the arrest of Jesus. While such a weapon clearly is a "makhaira" by ancient definition, the imprecise nature of the word as used in the New Testament cannot provide any conclusive answer.