The word covers two different things and the tradition itself distinguishes them. Martinezism is the theurgy of Martinez de Pasqually: elaborate ceremonial operations, performed at set times, aimed at obtaining a sensible sign that the operator's reconciliation with the divine was under way. Martinism proper is what Louis Claude de Saint Martin turned to after abandoning that: an interior way, without ceremony, pursued through prayer, reading and self-examination.
Papus's order, assembled in Paris in the 1880s, inherited both currents and gave them a small ritual frame: the mask, the cloak, and a short sequence of degrees conferred person to person. The doctrine underneath is the fall and the reintegration of humanity, which is Pasqually's frame carried into Saint Martin's method.
What followed is the field's standard outcome. The order fractured repeatedly through the twentieth century, through the deaths of its leaders, the two wars, and the occupation, and there are now many bodies using the name, several claiming the only valid transmission, in France, the United States and elsewhere. Untangling them is a specialist task and the atlas does not attempt it here.
The initiatic point is that Martinist transmission passes between individuals rather than through a temple, which is precisely what makes the lineages so hard to trace and so easy to assert.