Iirc, the fat Santa was a creation of Coca Cola. Most earlier depictions show him with normal weight.
No the first real written description for Santa were from “Twas the night before Christmas” originally titled “A visit from St. Nicholas”. In it we get the lines:
He had a broad face and a little round belly,
That shook, when he laughed like a bowlful of jelly.
He was chubby and plump, a right jolly old elf,
I wonder if Dickens’ description of the Ghost of Christmas Present influenced the idea of the “fat Santa”?