The likes of Robert Cutter, William Cromwell and Michal Farmer use the term three states, Rafe De Crespigny and Andrew Chittick on the other hand uses three kingdoms, depending on their translation of Chen Shou's Sanguozhi (often abbreviated to SGZ) ie Record of the Three Kingdoms/States. I'm not aware of anybody who uses the term three Empires.

I don't think the three states group will be making a major push and I certainly haven't heard of one. The term three kingdoms gives the general idea: civil war, three factions (well eventually and then it was four till 238 with the Gongsun clan of Liaodong) competing to unify the land. The term is well established, it is in modern media and of all the issues understanding the era faces, thanks to the novel shaping many perceptions, the choice of name is not one I have seen complained about.

Chen Shou was a proud man of Yi, a follower of the soothsayer Qiao Zhou, an officer of Shu and then, following Liu Shan's surrender, to Jin of the Sima clan and eventual unifiers of China. His SGZ, editing and compiling from the record departments of all three kingdoms into a collection of biographies, was a private work but his well-regarded work is the primary source (later added to by Liu Song scholar Pei Songzhi with annotations from other works about the three kingdoms and commentaries) for the era.

He did not choose to represent it as three Empires, his records did not give the full prestige and symbols of dynastic records: no annals (though ruler biographies acted for that), no treaties, chronological and genealogical tables not there. He used the Emperor names for the Wei dynasty but for Shu's two rulers, First and Second Sovereign (an honorific but one that also acted against their claims to be a continuation of the Han) and for Wu, he just called them by their names.

So why not three Empires by Chen Shou?

Chen Shou, as mentioned earlier, was an officer of Jin and though he has gained a reputation for remarkable neutrality, he had to navigate that world. Jin were the winners, so they had got the mandate of heaven, they were the legitimate dynasty. Jin had narrative needs (like not overly highlighting the whole awkward regicide in the open streets incident) including the journey of the mandate since they hadn't started the war as a faction. Jin's argument was it had gone from Han via Emperor Xian's abdication to Cao Pi and the Wei dynasty, the Cao family held it but had lost their way which justified the internal changes that saw the Sima's seized power in 249 and then Cao Huan abdicated to Sima Yan and Jin dynasty as Han had to Wei. Shu and Wu were rebels, their calendar systems were not to be followed, they had no mandate and their rulers had no right to claims to be the Son of Heaven.

Raising the rebels onto the same level as the dynasty Jin claimed their mandate (and their legitimacy) from would not have been a good way to safeguard his career or his life. While showing a degree of loyalty to his former masters in Shu and his beloved Yi was not frowned upon, there were limits to how far that could be pushed before it would raise questions about his loyalty. Raising Shu and Wu to states and carefully neutral wording in wars to avoid implying fights between them was emperor vs rebels was probably as far as he could go without risking major trouble.

I hope this was useful, let me know if you have more questions and have a lovely week ahead.