The figures and calculations on the share of animal agriculture in climate change differ widely. The most common and at the same time smallest figure (14.5%) comes from the Food and Agriculture Organization of the United Nations (FAO). Here it must be taken into account that the FAO is steered and financed by all global animal industries.
The following slides provide an overview of various analyses, their calculations and main issues.
A tabular one-page overview (PDF) incl. list of references and links to all references can be downloaded here:
Averting the climate catastrophe without a phase-out of animal agriculture is not possible, because not only must greenhouse gas emissions be reduced to zero, but also the CO2 - concentration in the atmosphere must decrease drastically.
A safe CO2 concentration in the atmosphere for the survival of mankind is 350 ppm . In 2020, the concentration was 413 ppm. To get back from there to the level of 350 ppm, a sequestration of 491 Gt CO2 is required . Technical solutions are lacking, so the only option is afforestation. Enormous areas are needed for this. A shift to a purely plant-based food system frees up an area of 31 million square kilometers . Renaturalization (reforestation, conversion back to peatlands, etc.) can sequester up to 547 Gt CO2  within 30 years and 810 Gt CO2  within 100 years (= carbon opportunity cost of animal agriculture).
The necessary sequestration of 491 Gt CO2can be achieved within 27 years.
An additional prerequisite for achieving the safe CO2 concentration is the reduction of CO2 emissions to net zero, which totaled 39 Gt CO2 per year in 2020. This essentially requires a complete phase-out of fossil energies and solutions for cement.
A sole reduction of CO2 - emissions to net zero does not solve the problem of high CO2 - concentrations in the atmosphere because CO2, unlike, for example, methane is a very long-lived gas over centuries.