This hadith is reported in Imam Ahmad's Musnad and also Sunan Abi Dawud. It is also reported that Imam Ahmad was asked about this hadith, and he commented:
That is, Abu Bakr has no powers to slay a man except for three reasons which the Messenger of Allah (ﷺ) had mentioned: disbelief after belief, fornication after marriage, or killing a man without (murdering) any man by him. The Prophet (ﷺ) had powers to kill. (Sunan Abi Dawud)
This explanation by Imam Ahmad is reported in Sunan Abi Dawud along with other books recording Imam Ahmad's statements.
To be clear, it is not about only the Prophet being able to judge. It is about the Prophet being the only one who can legislate who dies. We only follow what he legislated and judge based on that.
Killing people based on speaking harshly to Abu Bakr (RA) is not something the Prophet (SAW) legislated, so Abu Bakr (RA) says "That (deciding to kill someone for a new reason) is only for the Prophet (SAW)."
Also, note that insulting the Prophet may constitute disbelief in itself, so it may fall into the first category.