Is this hadith present in Sahih hadith books? (My Google search did not find it, but it could be due to English wording.)
This statement doesn't appear in any sahih book as it is even doubtful if it was a hadith at all. The only author who apparently attributed it to the prophet () was Burhan ad-Dyn al-Halabi (from Aleppo) علي بن برهان الدين الحلبي the author of as-Sirah al-Halabiyah (3/295) a biography of the prophet () where it is mentioned without any narrator chain as a response of our prophet () to the gifts of al-Muqawqis المقوقس the leader of the Coptic which included foodstuff.
The statement of this narration is:
In the following I will translate from Arabic these translation are of my own take them carefully!
نحن قوم لا نأكل حتى نجوع وإذا أكلنا لا نشبع
We are people who don't eat unless we are hungry and if we ate we don't get full up (with food).
It was apparently also quoted by ibn Qayyim in his Zaad al-Ma'aad and ibn Kathir in his al-Bidaya wa an-Nihayya. Al-Albani commented on this statement: It has no source it can't be truly attributed to the prophet () as far as we know. Nevertheless there's some truth in it and we can find some evidences supporting it like:
"I heard the Messenger of Allah (S.a.w) saying: 'The human does not fill any container that is worse than his stomach. It is sufficient for the son of Adam to eat what will support his back. If this is not possible, then a third for food, a third for drink, and third for his breath."
(Jami' at-Tirmidhi and also in Musnad Ahmad)
“I heard the Messenger of Allah (ﷺ) say: ‘A human being fills no worse vessel than his stomach. It is sufficient for a human being to eat a few mouthfuls to keep his spine straight. But if he must (fill it), then one third of food, one third for drink and one third for air.’” (Sunan ibn Majah)
And others that show the contrary:
...So I gave it to a man who drank his fill, then he gave it to another one, until it ended up with the Messenger of Allah (s.a.w), and all of the people had drank their fill. The Messenger of Allah (s.a.w) took the bowl, put it on his hand,then raised his head. He smiled and said: 'Abu Hurairah, drink.' So I drank, then he said: 'Drink.' I kept drinking and he kept on saying, 'Drink.' Then I said: 'By the One Who sent you with the truth! I have no more space for it.' So he took the bowl and praised Allah, mentioned His Name and drank.'”
(From a long hadith where all the ahl Sufa have been given to drink milk in Jami' at-Tirmidhi)
The hadith above shows that filling the stomach is allowed even if it is not the recommended. There are also narrations showing that occasionally the prophet () invited sahabah () to eat and they ate until they were full up with food (See Looking for a hadith about feeding 10 by 10 sahabah until they got full up?).
Nevertheless scholars consider a certain retention while eating as recommended ibn abi Zayd for example says in his ar-Rissala (with commentary -see here-):
The manners of eating include leaving a third of your stomach for food, a third for drink and a third for breath.
So when someone eats a lot of food, he has no space for breathing.
This is based on the hadith version from Sunan ibn Majah quoted above.
In a commentary of ibn abi Zayd's ar-Risala the author quoted the following:
مالك : ومن طب الأطباء أن ترفع يدك من الطعام وأنت تشتهيه
Imam Malik said: part of the medicine of physiscians is to raise your hand from a food you when you have ravenous appetite for it ...
وقال سحنون : كل شيء يعمل على الشبع إلا ابن آدم إذا شبع رقد
And Suhnun said: Every thing (creature) works after getting full up with food except the son of Adam, if he gets full up he gets sleepy
The wide spread statement saying "We don't eat until we hungry ... " cab hardly be attributed to the prophet () but it has some backup in the sunnah and is a good advice. But it is not a hadith let alone a sahih one.
My major source fatwa islamqa #115864 in Arabic.