The key to this problem lies in the meaning of "usurpation". The original intention of usurping the throne is that the courtiers seize the power of the monarch, which is what we often call usurping the throne. There are two meanings here, one is to be a courtier, and the other is to occupy the status of a monarch.
Moreover, usurpation also has an implied meaning. Generally speaking, usurpation will not cause too many internal wars and confrontations. Otherwise, it was not called usurpation in ancient times, but called counterattack. It's called a counterattack.
In history, class was called usurper. For example, Song Taizu Zhao Kuangyin's The Man in the Yellow Robe is a usurper.
Then compare the differences between Cao Pi and Liu Bang. Cao Pi, a courtier of the Han Dynasty, finally gained the throne through abdication without any major conflict, which can be called usurping the Han Dynasty, which accords with the definition of usurping the Han Dynasty. Therefore, there is no problem for Cao Pi to usurp the Han Dynasty.
Let's take a look at Liu Bang's situation. Although Liu Bang used to be a curator, he was a dirty official. Reluctant to call him a courtier of Qin, he will no longer be a courtier of Qin when he finally gains political power.
In addition, Liu Bang's second point is that Liu Bang's regime was not directly seized from the Qin Dynasty. Although Liu Bang entered Xianyang first, it was actually Xiang Yu, not Liu Bang, who had the greatest strength and power at that time. Liu bang did not seize power at that time. Then Liu Bang was named Hanwang by Xiang Yu instead, which shows that Liu Bang was subordinate to Xiang Yu at that time, and Liu Bang did not directly replace the regime of Qin Dynasty.
Later, Chu and Han fought for hegemony to establish the Han Dynasty, and it was Xiang Yu, not the Qin Dynasty, who competed with Liu Bang for political power. Therefore, for the Qin Dynasty, Liu Bang was only a rebellion, not a usurper, and he did not directly lead the Qin Dynasty.
A similar situation has happened in history. For example, Li Yuan of the Tang Dynasty was originally a courtier of the Sui Dynasty. He fought back and finally occupied the country. Generally speaking, the Sui Dynasty was founded by Li Yuan against the Sui Dynasty, not by Li Yuan, because he rose to fight for the world and experienced a large-scale war, which was generally called "not usurping the Sui Dynasty".
This comparison explains why Cao Pi is called a usurper, but Liu Bang is not a usurper of Qin.