Is it permissible for us to say that Husayn died as a martyr?

Yes, Husayn (may Allah be pleased with him) was martyred. He departed for Iraq after receiving letters from the people of Kufa inviting him to pledge allegiance, but they turned against him after Ubayd Allah ibn Ziyad took charge, and they killed his messenger, Muslim ibn Aqil. Despite the companions' advice not to go, Husayn insisted on proceeding. Upon reaching Karbala, and realizing the betrayal of the people of Kufa, he requested one of three things from the attacking army: to return to Mecca, or to go to Yazid, or to join the frontier garrisons for jihad. However, they refused unless he surrendered. When he refused, they fought and killed him unjustly as a martyr. His killing was one of the great calamities, and his status was elevated with Allah, who honored him with martyrdom so he could join his noble family.