Bukhari

Hadith Commentary:
Nawawi rahimahullah said that here "tamattu‘" refers to "qiran" (combining Hajj and ‘Umrah in one ihram). What happened was that at first the Prophet sallallahu alayhi wa sallam had assumed ihram only for Hajj, then he included ‘Umrah in it, and qiran is also called tamattu‘.
(Wahidi)

In this very hadith, mention is also made of the Prophet sallallahu alayhi wa sallam performing "raml" during the circumambulation (tawaf) of the Ka‘bah—
that is, walking briskly with the shoulders moving in a proud manner.
He did this in the first three circuits of tawaf, and in the remaining four he walked in the usual manner. This was done because the polytheists of Makkah had thought that the Muslims had become weak due to the fever of Madinah, so this action was first performed to prove their assumption wrong, and thereafter this practice remained a Sunnah forever.
(Wahidi)

There are many such historical commemorative acts in Hajj which are the memorials of the earlier elders, and for this reason they are considered among the rites (arkan) of Hajj, and lessons should be drawn from them. The act of raml is also such a historical act.

Hadith Commentary:
(1)
The purpose of Imam Bukhari rahimahullah in citing this hadith is to show that the Sunnah method regarding the sacrificial animal (qurbani) is to bring it from the "Hil" (outside the boundaries of the Haram) along with oneself.
If one purchases the sacrificial animal from within the Haram, then when going to ‘Arafah for Hajj, it is better to take it along. However, some scholars hold the view that the reason the Messenger of Allah sallallahu alayhi wa sallam brought the sacrificial animal from the Hil was because his house was outside the boundaries of the Haram; therefore, bringing the sacrificial animal from the Hil is not a Sunnah act in itself.
In any case, it is a Sunnah act to bring the sacrificial animal from the miqat or distant places, a practice which most people today have neglected.
(Fath al-Bari: 3/682) (2)
At this point, the term "tamattu‘" refers to Hajj Qiran, meaning that they availed themselves of both Hajj and ‘Umrah in a single journey.
In reality, the Messenger of Allah sallallahu alayhi wa sallam first entered into the state of ihram for Hajj, then included ‘Umrah within it.
From the perspective of language and meaning, it is permissible to apply the term "mutamatti‘" to a "qarin" (one performing Hajj Qiran), because in it, one benefits from a single miqat, a single ihram, and similar acts of Hajj.
In Hajj Tamattu‘, after performing ‘Umrah, one may exit the state of ihram and enjoy intimacy with one’s wife, whereas in Hajj Qiran, the benefit is in combining ‘Umrah with Hajj in a single ihram.
In light of other ahadith, this interpretation is absolutely necessary.
And Allah knows best.