All perfect praise be to Allaah, The Lord of the worlds. I testify that there is none worthy of worship except Allaah and that Muhammad is His slave and Messenger.
First of all, you should know that both ahaadeeth were reported by both Al-Bukhari and Muslim, and it is not that Al-Bukhari reported the first hadeeth and Muslim reported the second one, as is understood from your question.
The Prophet stopping at one place in order to search for the necklace does not necessarily mean that he did that by himself, but it could also mean that he stopped along with the people in order to find the necklace that had been lost.
Al-Baaji said in "Al-Muntaqa Sharh Muwatta’ Al-Imaam Maalik", “The saying of ‘Aa’ishah that the Prophet stopped to look for it; she wants to say by this that he stopped until he could look for it by the disappearance of darkness that prevents from searching for it or to wait for whomever he sent to look for it.”
It is for this reason that, in the hadeeth, it is mentioned that Abu Bakr As-Siddeeq reproachfully said to Aa'ishah, “You delayed the Prophet and the people.” This does not mean that all the people had gone in search for it.
Allaah knows best.