The short answer: the egg story appears in the Samguk Sagi too
The story of Jumong (朱蒙) born from an egg already appears in Book 13 of the Goguryeo Annals in the Samguk Sagi (三國史記). Yi Gyu-bo's Lay of King Dongmyeong (東明王篇) retells the same story as a long poem of 141 rhymes and 282 lines. For the King Gwanggaeto stele inscription, only one point can be confirmed: the Dongmyeong myth appears at its opening. How the stele's opening account diverges from the other two records cannot be confirmed from the sources used in this article.
The myth goes by another name as well. According to the Encyclopedia of Korean Culture, published by the Academy of Korean Studies, it is also called the Jumong myth (朱蒙神話), after its protagonist. This article uses Jumong throughout; the posthumous title the Samguk Sagi gives him is King Dongmyeongseong.
An egg about five doe in size: how the Goguryeo Annals tell it
Book 13 of the Goguryeo Annals in the Samguk Sagi describes Jumong's birth this way: sunlight shone upon Yuhwa (柳花), she conceived, and she gave birth to a single egg about five doe in size. The birth passage in the translation posted on Wikisource reads: "The sunlight followed her again and shone upon her, and because of this she conceived and bore a single egg, about five doe in size."
In the next scene King Geumwa throws the egg away. The Samguk Sagi records that he threw it to the dogs and the pigs, but none of them would eat it. It also records that he threw it into the middle of the road, and the oxen and horses stepped around it, and that he threw it out into the fields, and the birds covered it with their wings. The mother then wrapped the egg in something and kept it in a warm place, and a boy broke through the shell and came out. The Samguk Sagi calls the child's frame and features outstanding and remarkable.
The Encyclopedia of Korean Culture sums up the lineage briefly. Jumong's father is Haemosu (解慕漱), god of the sky; his mother is Yuhwa, daughter of Habaek (河伯), god of the river; and Jumong is born to them as an egg. The egg story is not an oddity confined to a single text but a frame that several records share.
The name's meaning and the founding in 37 BCE in the Samguk Sagi
The Samguk Sagi also explains the name: in the vernacular of Buyeo a skilled archer was called jumong, and the word became his name. The same book dates the founding and gives Jumong's age at the time as twenty-two. The year is the twenty-first year of Hyeokgeose, founder of Silla, a gapsin year — that is, 37 BCE.
Book 13 also describes the pursuit. Fish and soft-shelled turtles rose to the surface and formed a bridge, and Jumong crossed the river; the cavalry chasing him could not. The same book records the king's end briefly, in a Chinese original that reads: "秋九月,王升遐,時年四十歲。葬龍山,號東明聖王。" He died at the age of forty, was buried at Yongsan (龍山), and was called King Dongmyeongseong.
Readers who want to go to the original texts can check three places. The translation of Samguk Sagi Book 13 is available on Wikisource. The entries on the Dongmyeong myth and the Dongguk Yi Sangguk jip are in the Encyclopedia of Korean Culture, published by the Academy of Korean Studies, and the entry on the Jumong myth is in the Encyclopedia of Korean Folklore, published by the National Folk Museum of Korea.
The Lay of King Dongmyeong: 141 rhymes and 282 lines in Book 3 of the Dongguk Yi Sangguk jip, printed in 1241
The Lay of King Dongmyeong is carried in the Dongguk Yi Sangguk jip (東國李相國集), the collected poetry and prose of Yi Gyu-bo, a civil official of the late Goryeo period, which runs to 53 volumes in 13 books. His son Ham (涵) printed the first collection of 41 volumes in August 1241, and the later collection of 12 volumes followed in December of the same year. In 1251 his grandson Ikbae (益培) corrected and expanded the text at the Bunsa Daejang Dogam and issued it again. The edition that survives today is thought to be a recut made during the reign of King Yeongjo.
The Lay of King Dongmyeong appears in Book 3 of the first collection: 141 rhymes and 282 lines of five-syllable old-style verse. The Encyclopedia of Korean Culture explains that the work has the qualities that justify calling it a "heroic epic," and the same entry notes that it is the longest of the materials in this field.
That entry also lays out the poem's range. It opens with the meeting of Haemosu and Yuhwa, then follows Jumong's birth, his trials, and the founding of Goguryeo, and goes on to the succession of prince Yuri (類利) to the throne. It closes with a wish that the ruler govern the country with a generous and benevolent heart and with propriety. The preface also tells us that a history called the Gu Samguksa (舊三國史) once existed.
The Gu Samguksa line of transmission and the Samguk Sagi: point by point
The Encyclopedia of Korean Folklore, published by the National Folk Museum of Korea, weighs the records this way. Mythic accounts of Jumong appear in a number of texts, and among them the accounts in the Gu Samguksa and the Goguryeo Annals are judged the most detailed and the richest in mythic elements. The Dongmyeong myth also appears at the opening of the King Gwanggaeto stele inscription, though how that opening account diverges from the Samguk Sagi or the Lay of King Dongmyeong cannot be confirmed from the sources used in this article.
First, the egg comes from a different place. Samguk Sagi Book 13 says Yuhwa bore a single egg about five doe in size; the Gu Samguksa line of transmission says she bore a large egg from her left armpit. Second, the descriptions of his childhood differ. Samguk Sagi Book 13 says only that the child who broke out of the shell had an outstanding frame and appearance, while the Gu Samguksa line says Jumong spoke at one month old and shot flies with his bow without ever missing.
The episode of how he got his horse belongs to the Gu Samguksa line: Jumong drives a needle into the root of a fine horse's tongue to make it thin, then obtains that horse from King Geumwa. The endings diverge as well. Samguk Sagi Book 13 says the king died at forty and was buried at Yongsan, while the Gu Samguksa line says that in the nineteenth year of his reign he ascended to heaven and never came back down; the crown prince buried the jade whip the king had left behind at Yongsan and held the funeral rites there.
Putting it together
The Dongmyeong myth does not sit in a single text. The Encyclopedia of Korean Culture notes that the myth is carried in the Dongguk Yi Sangguk jip and also appears in the Samguk Yusa (三國遺事), the Samguk Sagi, the opening of the King Gwanggaeto stele inscription, and the Pyongyang section of the Dongguk yeoji seungnam, among others. The Northern Buyeo section of the Samguk Yusa calls the heavenly emperor of Northern Buyeo Haemosu and names his son Haeburu (解扶婁), then relates that Haeburu moved to Eastern Buyeo at the command of the Supreme Emperor and that Dongmyeong succeeded to Northern Buyeo.
The version of the story in wide circulation is not a straight transcription of any one text; it is explained as a blend of the accounts in the Samguk Sagi and the Samguk Yusa with the account in the Lay of King Dongmyeong. The Encyclopedia of Korean Folklore explains that the Jumong myth brings together the Buyeo-line Dongmyeong myth, the Hae-clan myth, the water-deity myth, and the Gojoseon myth. The Dangun gi (檀君記) describes Jumong and Buru as brothers by different mothers, and the Samguk Sagi says that when Yuhwa died in Eastern Buyeo, King Geumwa buried her with the rites due a queen dowager and built a shrine for her. It can therefore be said that readers are on safer ground if they note down the name of the text and the book number alongside whatever version of the story they are reading.