Would it be more historically correct or accurate to refer to the Timurids as a "Khanate" or an "Empire"?

I don't know the answer to your specific question, but I think a little bit of broader background might help. First, what's often translated as "emperor" is often really a title like "King of Kings", "Shah of Shahs", or "Khan of Khans", that is the person's authority is so high that they rule over other people who are generally though of as being the highest powers. The Khan of Khans equivalent is often written Khagan.

Second, we tend to think of titles as individual offices. Someone is president, or they are governor, or they are senator. In non-democratic states, titles often "stack". Thus, the full title of a late Ottoman Emperor was something along the lines of:

Sultan (given name) Khan, Sovereign of the House of Osman, Sultan of Sultans, Khan of Khans, Commander of the Faithful and Successor of the Prophet of the Lord of the Universe, Protector of the Holy Cities of Mecca, Medina and Jerusalem, Emperor of The Three Cities of Constantinople, Andrinopole and Bursa, and of the Cities of Damascus and Cairo, of all Azerbaijan, of the Magris, of Barka, of Kairuan, of Aleppo, of Arabic Iraq and of Ajim, of Basra, of El Hasa, of Dilen, of Raka, of Mosul, of Parthia, of Diyarbakır, of Cicilia, of the Vilayets of Erzurum, of Sivas, of Adana, of Karaman, Van, of Barbary, of Abyssinia, of Tunisia, of Tripoli, of Damascus, of Cyprus, of Rhodes, of Candia, of the Vilayet of the Morea, of the Marmara Sea, the Black Sea and also its coasts, of Anatolia, of Rumelia, Baghdad, Kurdistan, Greece, Turkistan, Tartary, Circassia, of the two regions of Kabarda, of Georgia, of the plain of Kypshak, of the whole country of the Tartars, of Kefa and of all the neighbouring countries, of Bosnia and its dependencies, of the City and Fort of Belgrade, of the Vilayet of Serbia, with all the castles, forts and cities, of all Albania, of all Iflak and Bogdania, as well as all the dependencies and borders, and many other countries and cities.

In English, we generally refer to the Ottoman Emperor as an emperor, but in Modern Turkish it's more common to refer to him as "Padişah". In Ottoman times, the emperor might be called Han (Khan), Padişah (Padishah), Sultan, or sometimes Kaiser (Caesar). Thus we see Turkic, Persian, Arabic, and Greco-Roman titles. Which one is the "real" title? Well, they're all real and all important (though over time I believe it became increasingly customary to give preference to one or two). I imagine something similar took place with the dynasties you're asking about. The old titles didn't disappear, but as new lands were conquered, local and translated titles accumulated.