You posted that link first ;) That graph is accompanied by descriptions, which are helpful in deciphering it.
> total agriculture contributions as 18.4% of total contributions and livestock at 5.8% total contributions
"Livestock & manure (5.8%): animals (mainly ruminants, such as cattle and sheep) produce greenhouse gases through a process called ‘enteric fermentation’ – when microbes in their digestive systems break down food, they produce methane as a by-product. This means beef and lamb tend to have a high carbon footprint, and eating less is an effective way to reduce the emissions of your diet."
"The overall picture you see from this diagram is that almost three-quarters of emissions come from energy use; almost one-fifth from agriculture and land use [this increases to one-quarter when we consider the food system as a whole – including processing, packaging, transport and retail]; and the remaining 8% from industry and waste."
So those 5.8% are just gases emited by those animals / from manure.
We'd have to add a part of land use, deforestation, wastewater, chemicals and energy to get the complete number.
The page was written in September 2020; we have more recent studies, I've posted one earlier, which gives the potential range for animal ag. between 16.5 and 28.1%.
https://www.researchgate.net/publication/352100490_Emissions...